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23  WEST  MAIN  STREET 

WEBSTER,  N.Y.  14580 

(716)  872-4503 


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CIHM/ICMH 

Microfiche 

Series. 


CIHM/ICMH 
Collection  de 
microfiches. 


Canadian  Institute  for  Historical  IVIicroreproductions  /  Institut  Canadian  de  microreproductions  historiques 


Technical  and  Bibliographic  Notes/Notes  techniques  et  bibliographiques 


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to 


The  Institute  has  attempted  to  obtain  the  best 
original  copy  available  for  filming.  Features  of  this 
copy  which  may  be  bibliographically  unique, 
which  may  alter  any  of  the  images  in  the 
reproduction,  or  which  may  significantly  change 
the  usual  method  of  filming,  are  checked  below. 


L'Institut  a  microfilm^  le  meilleur  exemplaire 
qu'il  lui  a  6x6  possible  de  se  procurer.  Les  details 
de  cet  exemplaire  qui  sont  peut-dtre  uniques  du 
point  de  vue  bibliographique,  qui  peuvent  modifier 
une  image  reproduite,  ou  qui  peuvent  exiger  une 
modification  dans  la  mdthode  normale  de  filmage 
sont  indiqu^s  ci-dessous. 


P« 
of 
fil 


D 


Coloured  covers/ 
Couverture  de  couleur 


I      I    Covers  damaged/ 


D 


D 


D 
D 


D 


Couverture  endommagde 


Covers  restored  and/or  laminated/ 
Couverture  restaur^  et/ou  pellicul6e 


I      I    Cover  title  missing/ 


Le  titre  de  couverture  manque 


I      I    Coloured  maps/ 


Cartes  gdographiques  en  couleur 

Coloured  ink  (i.e.  other  than  blue  or  black)/ 
Encre  de  couleur  (i.e.  autre  que  bleue  ou  noire) 


I      I    Coloured  plates  and/or  illustrations/ 


Planches  et/ou  illustrations  en  couleur 

Bound  with  other  material/ 
Reli6  avec  d'autras  documents 

Tiglit  binding  may  cause  shadows  or  distortion 
along  interior  margin/ 

La  re  liure  serr6e  peut  causer  de  I'ombre  ou  de  la 
distortion  le  long  de  la  marge  int^rieure 

Blank  leaves  added  during  restoration  may 
appear  within  the  text.  Whenever  possible,  these 
have  been  omitted  from  filming/ 
II  se  peut  que  certaines  pages  blanches  ajout^es 
lors  d'une  restauration  apparaissent  dans  le  texte, 
mais,  lorsque  cela  dtait  possible,  ces  pages  n'ont 
pas  6t6  filmies. 


D 
D 
D 
0 
0 
□ 
D 
D 
D 
D 


Coloured  pages/ 
Pages  de  couleur 

Pages  damaged/ 
Pages  endommagdes 

Pages  restored  and/or  laminated/ 
Pages  restaurdes  et/ou  pellicul^es 

Pages  discoloured,  stained  or  foxed/ 
Pages  d6color6es,  tachetdes  ou  piqu6es 

Pages  detached/ 
Pages  d^tachtes 

Showthrough/ 
Transparence 

Quality  of  print  varies/ 
Qualiti  in^gaie  de  I'impression 

Includes  supplementary  material/ 
Comprend  du  materiel  suppldmentaire 

Only  edition  available/ 
Seule  Edition  disponible 

Pages  wholly  or  partially  obscured  by  errata 
slips,  tissues,  etc.,  have  been  refilmed  to 
ensure  the  best  possible  image/ 
Les  pages  totalement  ou  partiellement 
obscurcies  par  un  feuillet  d'errata,  une  pelure, 
etc.,  ont  6ti  filmtes  d  nouveau  de  fapon  d 
obtenir  la  meilleure  image  possible. 


Oi 
be 
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fir 
sii 
or 


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Tl 
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M 
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Additional  comments:/ 
Commentaires  suppldmentaires; 


This  item  is  filmed  at  the  reduction  ratio  checked  below/ 

Ce  document  est  filmA  au  taux  de  reduction  indiquA  ci-dossous. 


10X 

14X 

18X 

22X 

26X 

XX 

> 

^ 

12X 

16X 

20X 

24X 

28X 

32X 

The  copy  filmed  here  has  been  reproduced  thanks 
to  the  generosity  of: 

D.  B.  Weldon  Library 
Univerilty  of  Western  Ontario 

The  images  appearing  here  are  the  best  quality 
possible  considering  the  condition  and  legibility 
of  the  original  copy  and  in  keeping  with  the 
filming  contract  specifications. 


L'exemplaire  film6  fut  reproduit  grAce  d  la 
g6n4rosit6  de: 

D.  B.  Weldon  Library 
University  of  Western  Ontario 

Les  images  suivantes  ont  6t6  reproduites  avec  le 
plus  grand  soin,  compte  tenu  de  la  condition  et 
de  la  nettetd  de  l'exemplaire  filmd,  et  en 
conformity  avec  les  conditions  du  contrat  de 
filmage. 


Original  copie&  in  printed  paper  covers  are  filmed 
beginning  with  the  front  cover  and  ending  on 
the  last  page  with  a  printed  or  illustrated  impres- 
sion, or  the  back  cover  when  appropriate.  All 
other  original  copies  are  filmed  beginning  on  the 
first  page  with  a  printed  or  illustrated  impres- 
sion, and  ending  on  the  last  page  with  a  printed 
or  illustrated  impression. 


The  last  recorded  frame  on  each  microfiche 
shall  contain  the  symbol  — ^  (meaning  "CON- 
TINUED"), or  the  symbol  V  (meaning  "END"), 
whichever  applies. 

Maps,  plates,  charts,  etc.,  may  be  filmed  at 
different  reduction  ratios.  Those  too  large  to  be 
entirely  included  in  one  exposure  are  filmed 
beginning  in  the  upper  left  hand  corner,  left  to 
right  and  top  to  bottom,  as  many  frames  as 
required.  The  following  diagrams  illustrate  the 
method: 


Les  exemplaires  originaux  dont  la  couverture  en 
papier  est  imprim6e  sont  film^s  en  commenpant 
par  le  premier  plat  et  en  terminant  soit  par  la 
dernidre  page  qui  comporte  une  empreinte 
d'impression  ou  d'illustration,  soit  par  le  second 
plat,  salon  le  cas.  Tous  les  autres  exemplaires 
originaux  sont  film6s  en  commenpant  par  la 
premidre  page  qui  comporte  une  empreinte 
d'impression  ou  d'illustration  et  en  terminant  par 
la  dernidre  page  qui  comporte  une  telle 
empreinte. 

Un  des  symboles  suivants  apparattra  sur  la 
dernidre  image  de  cheque  microfiche,  salon  le 
cas:  le  symbols  —^  signifie  "A  SUIVRE",  le 
symbols  V  signifie  "FIN". 

Les  cartes,  planches,  tableaux,  etc.,  peuvent  Atre 
film6s  d  des  taux  de  reduction  diffdrents. 
Lorsque  le  document  est  trop  grand  pour  dtre 
reproduit  en  un  seul  cliche,  il  est  film6  A  partir 
de  Tangle  sup^rieur  gauche,  de  gauche  d  droite, 
et  de  haut  en  bas.  en  prenant  le  nombre 
d'images  ndcessaire.  Les  diagrammas  suivants 
illustrent  la  mdthode. 


1 

2 

3 

1 

2 

3 

4 

5 

6 

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EDITED   BY 

WILLIAM  T.  HARRIS,  A.  M.,  LL.  D. 


Volume  XXX] II. 


hi 


:^jy 


;   W- 


THE 


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-i 


lyTEUXA  TIOXA  L   ED  VCA  TIOX  SEIUES 


THE 


PSYCHOLOGY  OF  NUMBEFi 

AND  ITS  APPLICATIOXS   TO 
METHODS   OF   TEACHING  ARITHMETIC 


BY 


JAMES  A.  McLELLAN,  A.  M.,  LL.  D. 

PKINCIPAL   UF   THK    UXTAUIO    SCHOOL   OF   PEDAGOGV,   TOROXTO 


AND 


JOHN  DEWEY,  Ph.D. 

HEAD   PROFESSOR   OF   PIIILOSOPHV    IX   THE    LNIVERSITY    OF   CHICAGO 


The  art  of  measuring  brings  the  world  into  subjection  to  man  ;  the  art 
of  wntHig  prevents  his  knowledge  from  perishing  along  with  himself-  to- 
gpthor  they  make  man-what  Nature  has  not  made  him-all-powerfuland 
eternal."-  MoMMSEN. 


NEW    YORK 
D.    APPLETON    AND    COMPANY 

1895 


\A<e>G>o 


4 


Copyright,  1895, 
By  D.   APPLETON  AND  C03IPANY. 


Electrotyped  and  Printed 

AT  THE  APPLETON  PbESS,  U.  S.  A. 


EDITOR'S  PREFACE. 


In  presenting  this  book  on  the  Psychology  of  Num- 
ber it  is  believed  that  a  special  want  is  supplied.  There 
is  no  subject  taught  in  th^' ^kjmpivtary  schools  that  taxes 
the  teacher's  resources  as  to  methods  and  devices  to  a 
greater  extent  than  arithmetic.  There  is  no  subject 
taught  that  is  more  dangerous  to  the  pupil  in  the  way 
of  deadening  his  mind  and  arresting  its  development,  if 
bad  methods  are  used.  The  mechanical  side  of  training 
must  be  joined  to  the  intellectual  in  such  a  form  as  to 
prevent  the  fixing  of  the  mind  in  thoughtless  habits. 
While  the  mere  processes  become  mechanical,  the  mind 
should  by  ever-deepening  insight  continually  increase 
its  powder  to  grasp  details  in  more  extensive  combina- 
tions. 

Methods  must  be  chosen  and  justified,  if  they  can 
be  justified  at  all,  on  psychological  grounds.  The  con- 
cept of  number  will  at  first  be  grasped  by  the  pupil  im- 
perfectly, lie  will  see  some  phases  of  it  and  neglect 
others.  Later  on  he  will  arrive  at  operations  w^hich 
demand  a  view  of  all  that  number  implies.  Each  and 
every  number  is  an  implied  ratio,  but  it  does  not  ex- 
press the  ratio  as  simple  number.  The  German  lan- 
guage is  fortunate  in  having  terms  that  cjxpress  the  two 
aspects  of  numerical  quantity.     Anzahl  expresses  the 


vi 


KhlToli'S   IMIKKACR 


multiplicity  uiul  Kinlwlt  the  unity.  Any  uuihIkm',  sjiy 
six,  fur  oxanipio,  has  these  t\v<j  {is[)eet,s :  it  is  a  iiiaiiit'oid 
of  units;  the  constituent  unit  whatever  it  is,  is  repeated 
six  times.  It  is  a  unitv  of  these,  and  as  such  nuiy  he  a 
constituent  unit  of  a  larg'cr  numher,  live  times  six,  for 
instance,  wherein  tlie  iive  represents  the  multiplicity 
{Anzahl)  and  the  six  the  constituent  unity  {Kin/wit).^' 

Xuniber  is  one  of  the  developments  of  quantity. 
,Its  multiplicity  and  unity  correspond  to  the  tw(>  more 
general  aspects  of  (piantity  in  general,  namely,  to  dis- 
creteness aud  continuity. 

There  is  such  a  thing  as  (pialitative  unity,  or  indi- 
viduality. Quantitative  unity,  unlike  individuality,  is 
always  divisible  into  constituent  units.  All  quantity  is 
a  unity  of  units.  It  is  composed  of  constituent  units, 
and  it  is  itself  a  constituent  unit  of  a  real  or  possible 
larger  unity.  Every  pound  contains  within  it  ounces  ; 
every  pound  is  a  constituent  unit  of  some  hundred- 
weight or  ton. 

The  simj)le  number  implies  both  })hases,  the  multi- 
plicity and  the  unity,  but  does  not  express  them  a<le- 
quately.  The  chikrs  thought  likewise  possesses  the 
same  inadequacy  ;  it  implies  more  than  it  explicitly 
states  or  holds  in  consciousness. 

This  twofoldness  of  number  becomes  explicit  in 
multiplication  and  division,  wdierein  one  number  is  the 
unit  and  the  other  expresses  the  multiplicity — the  times 
the  unit  is  taken.  Fractions  form  a  more  adequate  cx- 
presfeion  of  this  ratio,  and  require  a  higher  conscious- 
ness of  the  nature  of  quantity  than  sinifde  numbers  do. 
Hence  the  ditticulty  of  teaching  this  subject  in  the  ele- 


'•  riogol,  TiOgik,  B(l.  T,  1st  Th  ,  S.  225. 


KDITOirs    IMiKFACK. 


Yii 


meiitary  .school.  The  thought  of  }  (kMiuuicls  the  thought 
of  both  iiuin])CMs,  7  and  .s,  aiul  thu  thought  of  tlieir  luodi- 
Hciition  each  thi'ough  the  other. 

Tlie  methods  in  voi^nie  in  eleniontarv  K'hodls  are 
cliietly  hased  on  the  idea  that  it  is  necessary  to  ehnii- 
nate  the  ratio  idea  l)y  changing  one  of  the  terms  of  the 
fraction  to  a  quaHtative  unit  and  bv  this  to  chantre  the 
tliouglit  to  that  of  a  simi)le  numher.  Thus  lialves  and 
quarters  and  cents  and  dimes  are  tliouylit  as  individual 
things,  and  the  fractional  idea  suppressed. 

In  the  differential  calculus  ratio  is  must  adcfjuately 
expressed  as  the  fundamental  and  true  form  of  all 
quantity,  numher  included.  The  differential  of  w  and 
the  dili'erential  of  1/  are  ratios. 

The  authors  of  this  book  have  presented  in  an 
a(hnirable  manner  this  psychological  view  of  number, 
and  shown  its  application  to  the  correct  methods  of 
teaching  the  several  arithmetical  processes.  The  short- 
comings of  the  "fixed-unit"  theorv  are  traced  out  in 
all  their  consequences.  The  defects  of  a  view  which 
,  makes  unity  a  quaHtative  instead  of  a  quantitative  idea 
are  sure  to  a])pear  in  the  methods  of  solution  adopted. 

Fui)ils  studying  music  by  the  highest  method  learn 
thoroughly  those  combinations  which  involve  double 
counterpoint.  As  soon  as  the  hands  are  trained  to 
readily  execute  such  exercises  the  pupil  can  take  up  a 
sonata  of  .l>eethoven  or  a  fugue  of  J3ach,  and  soon  be- 
come familiar  with  it.  (^n  the  plan  of  the  old  lessons 
in  counterpoint,  the  pupil  found  himself  helpless  before 
such  a  composition.  His  phrases  furnished  110  key  to 
the  compositions  of  Bach  or  Beethoven,  because  the 
latter  are  constructed  on  a  different  counterpoint. 


VIU 


EDITOR'S  PRKFACK. 


So  the  methods  of  teaching  arithmetic  by  a  '^  fixed- 
unit  "  system  do  not  lend  towards  the  higlier  mathe- 
matics, but  away  from  it.  They  furnish  little,  if  any, 
training  in  thinking  the  ratio  invoU'ed  in  the  very  idea 
of  number. 

The  psychology  of  number  recjuires  that  the  meth- 
ods be  chosen  with  reference  to  their  power  to  train 
the  mind  of  the  pupil  into  this  consciousness  of  the 
ratio  idea.  The  steps  should  be  short  and  the  ascent 
gradual ;  but  it  should  be  continuous,  so  that  the  pupil 
constantly  gains  in  his  ability  to  hold  in  consciousness 
the  unity  of  the  two  aspects  of  quantity,  the  unity  of 
the  discrete  and  the  continuous,  the  unity  of  the  nmlti- 
plex  and  the  simple  unit. 

Measurement  is  a  process  that  makes  these  elements 
clear.  The  constituent  unit  becomes  the  including  unit, 
and  vice  versa,  throuo-h  beino;  measured  and  beinc:  made 
the  measure  of  others.  This,  too,  is  involved  in  using 
the  decimal  system  of  numeration,  and  in  understand- 
ing the  different  orders  of  units,  each  of  which  both  in- 
cludes constituent  units  and  is  included  as  a  constituent 
of  a  higher  unit. 

The  hint  is  obtained  from  this  that  the  first  lessons 
in  arithmetic  should  be  based  on  the  practice  of  meas- 
uring in  its  varied  applications. 

Again,  since  ratio  is  the  fundamental  idea,  one  sees 
how  fallacious  are  those  theories  which  seek  to  lay  a 
basis  for  mathematics  by  at  first  producing  a  clear  and 
vivid  idea  of  unity — as  though  the  idea  of  quantity 
were  to  be  built  up  on  this  idea.  It  is  shown  that  sucli 
abstract  unit  is  not  yet  quantity  nor  an  element  of  (jujin- 
tity,  but  sinqily  the  idea  of  individuality,  which  is  still 


EDITOR'S   PIIKFACE. 


IX 


a  qualitative  idea,  and  does  not  become  quantitative 
until  it  is  conceived  as  composite  and  made  up  of  con- 
stituent units  homogeneous  with  itself. 

The   true  psychological  theory  of   number  is   the 
panacea  for  that   exaggeration   of   the   importance  of 
arithmetic  which  prevails  in   our  elementary  schools. 
As  if  it  were  not  enough  that  the  science  of  number  is 
indispensable  for  the  con(|uest  of  Nature  in  time  and 
space,  these  qualitative-unit  teachers  make  the  mistake 
of  supposing  that  arithmetic  deals  with  spiritual  being 
as  much  as  with  matter  ;  they  confound  quality  with 
quantity,   and    conse(iuently   mathematics   with   meta- 
physics.     Mental   arithmetic    becomes    in    their   psy- 1 
chology  "  the  discipline  for  the  pure  reason,"  although  ) 
as  a  matter  of  fact  the  three  figures  of   the  regular  \ 
syllogism  are  neither  of  them  employed  in  mathematical  j 


reasoning. 


W.  T.  Harris. 


Washington,  D.  C,  June  ;25,  1895. 


m 


f 


PEEFACE 


V 


It  is  perliaps  natural  that  a  growing  impatience 
with  the  meagre  results  of  the  time  given  to  arithmetic 
in  tlie  traditional  course  of  the  schools  should  result  in 
attacks  upon  that  study.  While  not  all  educational  ex- 
perts w^ould  agree  that  it  is  the  "  most  useless  of  all  sub- 
jects "  taught,  there  is  an  increasing  tendency  to  think 
of  it  and  speak  of  it  as  a  necessary  evil,  and  therefore 
to  be  kei)t  within  the  smallest  possible  bounds.  How- 
ever natural  this  reaction,  it  is  none  the  less  unwise 
when  turned  against  arithmetic  itself,  and  not  against 
stupid  and  stupefying  ways  of  teaching  it.  So  con- 
ceived, the  movement  stands  only  for  an  aimless  swincr 
of  the  scholastic  pendulum,  sure  to  be  followed  by  an 
equally  unreasonable  swing  to  the  other  extreme.  If 
methods  which  cut  across  the  natural  grain  of  the  men- 
tal structure  and  resist  the  straixditforward  workings 
of  the  mental  machinery,  waste  time ,  create  apathy  and 
disgust,  dull  the  ])ower  of  quick  peiception,  and  culti- 
vate habits  of  inaccurate  and  disconnected  attention, 
what  occasion  for  surprise  ?  Because  wrong  methods 
breed  bad  results,  it  hardly  follows  that  education  can 
be  made  synnnetrical  by  omitting  a  subject  which 
stands  par  eirrellence  for  clear  and  clean  cut  methods 

xi 


Xll 


PREFACE. 


of  tlioiiglit,  wliicli  forms  the  introduction  to  all  eilective 
interpretation  of  >y"ature,  and  is  a  powerful  instrument 
in  the  reij^ulation  of  social  intercourse. 

It  is  custi)marj  now  to  divide  studies  into  "form" 
studies  and  '"content"  studies,  and  to  de])reciate  arith- 
metic on  the  ground  that  it  is  merely  formal.  But  how 
are  we  to  sej)arate  form  and  content,  and  regard  one  as 
good  in  itself  and  the  other  as,  at  best,  a  necessaiy  evil  ? 
If  we  may  para})lirase  a  celebrated  saying  of  Kant's, 
while  form  without  content  is  bai'ren,  content  without 
form  is  nnishv.  An  education  which  ne<»:lects  the  for- 
mal  reIationshi])s  constituting  the  framework  of  the 
subject-matter  taught  is  inert  and  supine.  The  ])eda- 
gogicid  problem  is  not  solved  by  railing  at  "  form,"'  but 
in  discovering  what  kind  of  form  we  are  dealing  with, 
how  it  is  related  to  its  own  content,  and  in  work- 
ing out  the  educational  methods  which  answer  to  this 
relationship.  Because,  in  the  case  of  number,  "  form  " 
represents  the  measured  adjustment  of  means  to  an  end, 
the  rhythmical  balancing  of  parts  in  a  whole,  the  mas- 
tery of  lorni  represents  directness,  accuracy,  and  econ- 
omy of  ])erce]ition.  the  power  to  discriminate  the  rele- 
vant from  the  irrelevant,  and  ability  to  mass  and  con- 
verge relevant  material  upon  a  destined  end — repre- 
sents, in  short,  precisely  what  we  understand  by  good 
sense,  by  good  judgment,  the  power  to  put  two  and  two 
together.  AVhen  taught  as  this  sort  of  form,  arithmetic 
aft'ords  in  its  own  j)lace  an  unrivalled  means  of  men- 
tal disc'  ^''le.  It  is,  perhaps,  more  than  a  coincidence 
that  til  ^articular  school  of  educational  thought  which 
is  most  active  in  urging  the  merely  '' formaP'  quality 
of  arithmetic  is  also  the  one  which  stands  most  sys- 


w 


PJiEFACE. 


Xlll 


teiiiatieally  for  what  is  coiiflcinned   in    the  following 
pages  as  the  "fixed  unit"  method  of  teaching. 

As  foi*  the  counterpart  objection  that  nnmber  work 
is  lacking  in  ethical  substance  and  stimulus,  much  may 
be  learned  from  a  study  of  Greek  civilization,  from  the 
recognition  of  the  part  which  Greek  theory  and  prac- 
tice assigned  to  the  ideas  of  rhythm,  of  balance,  of 
measure,  in  moral  and  aesthetic  culture.  That  the 
Greeks  also  kept  their  arithmetical  training  in  closest 
connection  with  the  study  of  spatial  forms,  with  meas- 
urement, may  again  be  more  than  a  coincidence.  Even 
u])on  its  merely  formal  side,  a  study  which  requires 
exactitude,  continuity,  patience,  which  automatically  re- 
jects all  falsification  of  data,  all  slovenly  manipulation, 
which  sets  u^  a  controlling  standard  of  balance  at  every 
point,  cai-  ...dly  be  condemned  as  lacking  in  the  ethical 
element.  But  this  idea  of  balance,  of  compensation,  is 
more  than  formal.  Xumber  represents,  as  is  shown  in 
the  following  pages,  vdhiatioii ;  nuTiber  is  the  tool 
wherein'  modern  society  in  its  vast  and  intricate  pro- 
cesses of  exchange  introduces  system,  balance  and  econ- 
omy into  those  relationships  upon  which  our  daily  life 
depends.  Properly  conceived  and  presented,  neither 
geography  nor  history  is  a  more  effective  mode  of 
bj'inging  home  to  the  ])upil  the  realities  of  the  social 
environment  in  which  he  lives  than  is  arithmetic.  So- 
ciety has  its  form  also,  and  it  is  found  in  the  processes 
of  fixing  standards  of  value  and  methods  of  valuation, 
the  ])rocesses  of  weighing  and  counting,  whether  dis- 
tance, size,  or  quality  ;  of  measuring  and  fixing  bounds, 
whether  in  space  or  time,  and  of  balancing  the  various 
resulting  values  against  one  anotlier.     Arithmetic  can 


XIV 


PREFACE. 


not  be  properly  taught  without  being  an  introduction 

to  this  form. 

Thanks  are  due  to  Mr.  AYilliam  Scott,  of  tho 
Toronto  Normal  School,  for  some  assistance  with  tlic 
proofs ;  and  to  ^Ir.  Alfred  T.  De  Lury,  of  University 
College,  Lecturer  on  Methods  in  Mathematics  in  Ontario 
School  of  Pedagogy,  for  valued  practical  assistance. 

Aiigitst  13,  IS'Jo. 


nsRs 


CONTENTS. 


Editor's  Preface 
Author's  Preface 


PAGE 

V 

xi 


CHAPTER 

l._\ViiAT  Psychology  can  do  for  the  Ifacher    . 
1I._The  Psychical  Nature  of  Numrfr    .... 
in.— The  Origix  of  Number  :  Dependence  of  Number  on 
Measurement,  and  of  Measurement  on  Adjust- 
ment of  Activity 

IV._The  Origin  of  Number  :    Summary   and   Applica- 
tions   

v.— The  Definition,  Aspects,  and  Factors  of  Numerical 

Ideas  

VI.— The  Development  of  Number;  or,  the  Arithmetical 

Operations 

VII.— Numerical  Operations  as  External  and  as  Intrin- 
sic to  Number  

Vin.— On  Primary  Number  Teaching 

IX.— On  Primary  Number  Teaching 

X.~NoTATioN,  Addition,  Subtraction        .        .        .        . 
XI.— Multiplication  and  Division      .        .        .        •        • 

XII. —Measures  and  Multiples 

[XIII. — Fractions 

XIV.—Decimals 

XV.— Percentage  and  its  Applications       .        .        .        . 

XVI.— Evolution 

2 


1 
23 


35 

52 

08 

93 

119 
144 
1G6 
190 
207 
227 
241 
261 
279 
297 


THE   PSYCHOLOGY  OF  NUMBER. 


CHAPTER   I. 

WHAT    PSYCHOLOGY    CAN    DO    FOR    THE    TP:ACHER. 

The  value  of  any  fcKit  or  theory  as  bearing  on  hu- 
man activity  is,  in  tl)e  long  run,  determined  by  practi- 
cal application — that  is,  by  using  it  for  accomplishing 
some  definite  purpose.  If  it  works  well — if  it  removes 
friction,  frees  activity,  economizes  effort,  makes  for 
richer  results — it  is  valuable  as  contributing  to  a  perfect 
adjustment  of  means  to  end.  If  it  makes  no  such  con- 
tribution it  is  practically  useless,  no  matter  what  claims 
may  be  theoretically  urged  in  its  behalf.  To  this  the 
question  of  the  relation  between  psychology  and  educa- 
tion presents  no  exception.  The  value  of  a  knowledge 
of  psychology  in  general,  or  of  the  psychology  of  a  par- 
ticular subject,  will  be  best  made  known  by  its  fruits. 
No  amount  of  argument  can  settle  the  question  once  for 
all  and  in  advance  of  any  experimental  work.  But, 
since  education  is  a  rational  process,  that  is  a  process  in 
harmony  with  the  laws  of  psychical  development,  it  is 
plain  that  the  educator  need  not  and  should  not  depend 
upon   vague  inductions  from  a  practice  not  grounded 

upon  principles.     Psychology  can  not  dispense  with  ex- 

i 


TIIK    l\SVCllOLO(iY   OF   NUMUKR. 


])(.'riciicc,  nor  can  experience,  if  it  is  to  be  rational,  dis- 
pense with  psycliolo^^y.  It  is  ])ossil)lo  to  make  actual 
practice  less  a  matter  of  mere  experiment  and  more  a 
matter  of  I'cason  ;  to  make  it  contribute  directly  and 
economically  to  a  rich  and  rij)e,  because  rational,  experi- 
ence. And  this  the  educjitional  psychologist  attempts 
to  do  by  indicating  in  what  directions  help  is  likely  to 
be  found  ;  by  indicating  what  kind  of  psychology  is 
likely  to  hel[)  and  what  is  not  likely;  and,  finally,  by 
indicating  what  valid  reasons  there  are  for  anticipating 
any  help  at  all. 

I.  As  to  the  last  point  suggested,  that  psychology 
ought  to  help  the  educator,  there  can  be  no  disagreement. 
In  i\\Q,Jiri<t  place  the  study  of  psychology  has  a  high  dis' 
cij//huf/'(/  value  for  the  teacher.  It  develops  the  power 
of  connected  thinkintic  and  trains  to  lo<^ical  habits  of 
mind.  These  qualities,  essential  though  they  are  in 
thorough  teaching,  there  is  a  tendency  to  undervalue 
in  educational  methods  of  the  present  time  when  so 
much  is  made  of  the  accumulation  of  facts  and  so  little 
of  their  organization.  In  our  eager  advocacy  of  "facts 
and  things'"  we  are  apparently  forgetting  that  these  are 
comparatively  worthless,  either  as  stored  knowledge  or 
for  developing  jiower,  till  they  have  been  subjected  to  the 
discriminating  and  formative  energy  of  the  intelligence. 
Unrelated  facts  are  not  knowledge  any  more  than  the 
words  of  a  dictionai'v  are  connected  thoue-hts.  And  so 
the  work  of  getting  "things"  maybe  carried  to  such 
an  extent  as  to  burden  the  mind  and  check  the  growth 
of  its  higher  powers.  There  may  be  a  surfeit  of  things 
with  the  usual  consequence  of  an  impaired  mental  di- 
gestion.    It  is  pretty  generally  conceded  that  the  num- 


WHAT  rSYCIIOLOGV  CAN  DO  FOH  THE  TEACHER.     3 


ber  of  facts  nioinorized  is  by  no  means  a  measure  of  the 
amount  of  ])owei'  developed  ;  indeed,  unless  reilection 
has  been  exercised  step  by  step  with  observation,  the 
mass  of  power  gained  may  turn  out  to  be  inversely  })ro- 
portiunal  to  the  multitude  of  facts.  This  does  not  mean 
that  there  is  any  opposition  between  reilection  and 
true  observation.  There  can  not  be  observation  in  the 
best  sense  of  the  word  without  reHection,  nor  can  re- 
flection fail  to  be  an  eti'ective  preparation  for  obser- 
vation. 

It  will  be  readily  admitted  that  this  tendency  to 
exalt  facts  unduly  may  be  checked  by  the  study  of  psy- 
chology. Here,  in  a  comparatively  absti'act  science, 
there  miust  be  reilection — abstraction  and  generalization. 
In  nature  study  we  gather  the  facts,  and  we  may  reflect 
upon  the  facts  :  in  mind  study  we  must  reflect  in  order 
to  get  the  facts.  To  observe  the  subtle  and  complex 
facts  of  mind,  to  discriminate  the  elements  of  a  con- 
sciousness never  the  same  for  tw^o  successive  moments, 
to  give  unity  of  meaning  to  these  abstract  mental  phe- 
nomena, demands  such  concentration  of  attention  as 
must  secure  the  growth  of  mental  power — power  to 
master,  and  not  be  mastered  by,  the  facts  and  ideas  of 
whatever  kind  which  may  be  crowding  in  upon  the 
mind  ;  to  resolve  a  complex  subject  into  its  component 
parts,  seizing  upon  the  most  important  and  holding 
them  clearly  deflned  and  related  in  consciousness ;  to 
take,  in  a  word,  any  "  chaos  "  of  experience  and  reduce 
it  to  harmony  and  system.  This  analytic  and  relating 
power,  which  is  an  essential  mark  of  the  clear  thinker, 
is  the  prime  qualification  of  the  clear  teacher. 

But,  in  the  second  place,  the  study  of  psychology  is 


4  TIIH    I»SVC"Il()L()(iY   OF   XL'MHKR. 

of  still  more  vuliie  to  the  teiicliur  in  its  bearing  u])on  his 
pmctlcal  or  strictly  j)rofes8ionul  tniinin<j^. 

Every  one  grunts  tluit  the  j)rinuiry  uini  of  edueution 
is  the  training  of  the  powers  of  intelligence  and  will — 
that  the  object  to  be  attained  is  a  certain  (quality  of 
character.  To  say  that  the  purpose  of  education  is 
"an  increase  of  the  [)owers  of  the  mind  rather  than 
an  enlargement  of  its  ])ossessions  "  ;  that  education  is  a 
science,  the  science  of  the  formation  of  character ;  that 
character  means  a  measure  of  mental  power,  nuistery  of 
truths  and  laws,  love  of  beauty  in  nature  and  in  art, 
strong  liuman  sympathy,  and  unswerving  moral  recti- 
tude ;  that  the  teacher  is  a  trainer  of  mind,  a  former  of 
character ;  that  he  is  an  artist  above  nature,  vet  in  bar- 
mony  with  nature,  who  applies  the  science  of  education 
to  help  another  to  the  full  realization  of  his  personality 
in  a  character  of  strength,  beauty,  freedom — to  say  tliis 
is  simply  to  proclaim  that  the  problem  of  education  is 
essentially  an  ethical  and  psychological  problem.  This 
problem  can  be  solved  only  as  we  know  the  true  nature 
and  destination  of  man  as  a  rational  being,  and  the  ra- 
tional methods  by  which  the  perfection  of  his  nature 
may  be  realized.  Every  aim  proposed  by  the  educator 
which  is  not  in  harmony  with  the  intrinsic  aim  of  hu- 
man nature  itself,  every  method  or  device  em{)loyed 
by  the  teacher  that  is  not  in  perfect  accord  with  the 
mind's  own  workings,  not  only  wastes  time  and  energy, 
but  results  in  positive  and  permanent  harm  ;  running 
counter  to  the  true  activities  of  the  mind,  it  certainly 
distorts  and  may  possibly  destroy  them.  To  the  edu- 
cator, therefore,  the  only  solid  ground  of  assurance  that 
he  is  not  setting  up  impossible  or  artificial  aims,  that  he 


1 
I 


WHAT  PSYCIIOLOCiV  CAN   DO  FOR  THE  TEACIIEU. 


5 


is  not  usiiif^  inelYoctivc  and  pcrvertiiiii^  metliods,  is  a 
clear  and  deiinite  kiio\vled<^e  of  the  normal  end  and 
the  normal  forms  of  mental  action.  To  ki.ow  these 
thini»;s  is  to  be  a  true  psychologist  and  a  true  moralist, 
and  to  have  the  essential  (pialitications  of  the  true  edu- 
cationist. ]]rietly,  only  psychology  and  ethics  can  take 
education  out  of  its  purely  empirical  and  rule-of-thumb 
stau:e.  flust  as  a  knowledi»:e  of  nuithematics  and  me- 
chanics  has  wrought  marvelous  improvements  in  all  the 
arts  of  construction  ;  just  as  a  knowledge  of  steam  and 
electricity  has  made  a  revolution  in  modes  of  connnu- 
nication,  travel,  and  transportation  of  commodities  ; 
just  as  a  knowledge  of  anatomy,  physiology,  patholo- 
gy has  transformed  medicine  from  empiricism  to  ap- 
plied science,  so  a  knowledge  of  the  structure  and 
functions  of  the  human  being  can  alone  elevate  the 
school  from  the  position  of  a  mere  workshop,  a  more 
or  less  cumbrous,  uncertain,  and  even  baneful  institu- 
tion, to  that  of  a  vital,  certain,  and  effective  instrument 
in  the  greatest  of  all  constructions— the  building  of  a 
free  and  powerful  character. 

Without  the  assured  methods  and  results  of  science 
there  are  just  three  resoui-ces  available  in  the  work  of 
education. 

1.  The  "first  is  native  tact  and  skilly  the  intuitive 
power  that  comes  mainly  from  sympathy.  For  this 
personal  power  there  is  absolutely  no  substitute.  "  Any 
one  can  keep  school,''  perhaps,  but  not  every  one  can 
teach  school  any  more  than  every  one  can  become  a 
capable  painter,  or  an  able  engineer,  or  a  skilled  artist 
in  any  direction.  To  ignore  native  aptitude,  and  to  de- 
pend wholly,  or  even  chiefly,  upon  the  knowledge  and 


6 


THE  PSYCHOLOGY  OF  NUMBER. 


use  of  "  methods,"  is  an  error  fatal  to  the  best  interests 
of  education  ;  and  there  can  be  no  question  that  many 
schools  are  sufterinij:  frii-'htfuUv  from  iij^noring^  or  un- 
dervaluing  this  paramount  (|ualitication  of  the  true 
teacher.  But  in  urging  the  need  of  ps3'chology  in 
the  preparation  of  the  teacher  there  is  no  question  of 
ignoring  personal  power  or  of  finding  a  substitute  for 
personal  magnetism.  It  is  only  a  (piestion  of  provid- 
ing the  best  opportunities  for  the  exereit>e  of  native 
capacity — for  the  fullest  develo])ment  and  most  fruitful 
application  of  endowments  of  heart  and  brain.  Train- 
ing and  native  outfit,  culture  and  nature,  are  never  op- 
posed to  eacii  other.  It  is  always  a  question,  not  of 
suppressing  or  superseding,  l)ut  of  cuhivating  native 
instinct,  of  training  natural  ecpiipment  to  its  ripest  de- 
velopment and  its  richest  use.  A  Pheidias  does  not 
despise  learning  the  principles  necessary  to  the  mastery 
of  his  art,  nor  a  Beethoven  disregard  the  knowledge 
requisite  for  the  complete  technical  skill  through  which 
he  gives  expression  to  his  genius.  In  a  sense  it  is  true 
that  the  great  artist  is  born,  not  made ;  but  it  is  ecpially 
true  that  a  scientific  insight  into  the  technics  of  his  art 
heljys  to  make  him.  And  so  it  is  with  the  artist  teacher. 
The  greater  and  more  scientilic  his  knowleduje  of  human 
nature,  the  more  ready  and  skilful  will  be  his  ap])li- 
cation  of  principles  to  varying  circumstances,  and  the 
larger  and  more  perfect  will  be  t\n}.  })roduct  of  his 
artii^tic  skill. 

But  the  ixenius  in  education  is  as  rare  as  the  irenius 
in  other  realms  of  human  activitv.  Education  is,  and 
forever  will  be,  in  the  hands  of  ordinary  men  and 
women;   and  if  psychology — as  the  basis  of  scientific 


WHAT  PSYCHOLOGY  CAN  DO  FOR  THE  TFJACIIER.     7 


iiibiglit  into  Inuiian  nature — is  of  liigli  value  to  tlie  few 
Avlio  possess  genius,  it  is  indispensable  to  the  many  wlio 
Lave  not  genius.  Fortunately  for  the  race,  most  }ier- 
sons,  though  not  "  born  "  teachers,  are  endowed  with 
some  ''genial  impulse,"  some  native  instinct  and  skill 
for  education  ;  for  the  cardinal  requisite  in  this  en- 
dowment is,  after  all,  sympathy  with  human  life  and  its 
aspirations.  We  are  all  born  to  be  educators,  to  be 
parents,  as  we  are  not  born  to  be  engineers,  or  sculp- 
tors, or  musicians,  or  painters.  J^ative  capacity  for 
education  is  therefore  much  more  common  than  native 
capacity  for  any  other  calling.  Were  it  not  so,  human 
society  could  not  hold  together  at  all.  But  in  most 
people  this  native  sympathy  is  either  dormant  or  blind 
and  irregular  in  its  action ;  it  needs  to  be  awakened,  to 
be  cultivated,  and  al)ovo  all  to  be  intelligently  directed. 
The  instinct  to  walk,  to  speak,  and  the  like  are  imperi- 
ous instincts,  and  yet  they  are  not  wholly  left  to  "  na- 
ture "  ;  we  do  not  assume  that  they  will  take  care  of 
themselves  ;  we  stimulate  and  guide,  we  su]">ply  them 
with  j)roper  conditions  and  material  for  their  develop- 
ment. 80  it  must  be  with  this  instinct,  so  common  yet 
at  present  so  comparatively  inelfectiv^e,  which  lies  at 
the  heart  of  all  educational  efforts,  the  instinct  to  help 
others  in  their  struggle  for  self-mastery  and  self-expres- 
sion. The  very  fact  that  this  instinct  is  so  strong,  and 
all  but  universal,  and  that  the  happiness  of  the  individ- 
ual and  of  the  race  so  largely  depends  upon  its  develop- 
ment and  intelligent  guidance,  gives  greater  force  to 
the  demand  that  its  growth  may  be  fostered  by  favour- 
able conditions ;  and  that  it  may  be  made  certain  and 
reasonable  in  its  action,  instead  of  being  left  blind  and 


w 


8 


TlIK    PSYCHOLOGY   OF   NUMBER. 


falteriiis:,  as  it  surely  Avill  be  with  out  rational  culti- 
vation. 

To  this  it  luav  be  added  that  native  endowment  can 
work  itself  out  in  the  best  i)ossible  results  only  when 
it  works  under  riu'ht  conditions.  Even  if  scientiiic  in- 
sight  were  not  a  necessity  for  the  true  educator  himself, 
it  would  still  remain  a  necessity  for  others  in  oi'der 
that  they  might  not  obstruct  and  possibly  drive  from 
the  profession  the  teacher  possessed  of  the  iid)orn 
divine  light,  and  restrict  or  ])aralyze  the  efforts  of  the 
teacher  less  richly  endowed.  It  is  the  mediocre  and 
the  buuii^ler  who  can  most  readilv  accommodate  himself 
to  the  conditions  imposed  by  ignorance  and  routine;  it 
is  the  higher  type  of  mind  and  heart  which  suffers  most 
from  its  encounter  with  incapacity  and  ignorance. 
One  of  the  greatest  hindrances  to  true  educational 
progress  is  the  reluctance  of  the  best  class  of  minds  to 
en<];:a<>:e  in  educational  work  precisely  because  the  ii'en- 
eral  standard  of  ethical  and  psvcholoii'ical  knowledire  is 
so  low  that  too  often  high  id(nils  are  belittled  and  efforts 
to  realize  them  even  vigorously  opposinl.  The  educa- 
tional genius,  the  earnest  teacher  of  any  class,  has  little 
to  expect  from  an  indifference,  or  a  stolidity,  which  is 
proof  alike  against  the  facts  of  experience  and  the  dem- 
onstrations of  science. 

2.  The  Secoiul  Resource  is  {'.vj}en'erwe.  This,  again, 
is  necessary.  Psvcholoirv  is  not  a  short  and  easy  path 
that  renders  personal  experience  su])erliuous.  The  real 
question  is :  AVhat  kind  of  experience  shall  it  be  ?  It 
is  in  a  way  perfectly  true  that  only  by  teachiuir  can  one 
become  a  teacher.  J>ut  not  any  and  every  sort  of  thiuir 
which  passes  for  teaching  or  for   "experience"   will 


WHAT  PSYCHOLOGY  CAN  DO  FOR  THE  TEACHER.      0 

make  a  teaclier  any  more  than  .dimply  sawing  a  buw 
across  violin  strings  will  make  a  violinist.  It  is  a  cer- 
tain quality  of  practice,  not  mere  practice,  which  pro- 
duces the  expert  and  the  artist.  Unless  the  practice  is 
based  upon  rational  princi})les,  upon  insight  into  facts 
and  their  meaning,  "  experience  "  simply  fixes  incorrect 
acts  into  wrong  habits.  Nonscientific  practice,  even  if 
it  iinally  reaches  sane  and  reasonable  results — which  is 
very  unlikely — does  so  by  unnecessarily  long  and  cir- 
cuitous routes ;  time  and  enera;v  are  wasted  that  might 
easily  be  saved  by  wise  insight  and  direction  at  the 
outset. 

The  worst  thing  about  empiricism  in  every  depart- 
ment of  human  activity  is  that  it  leads  to  a  blind  ob- 
servance of  rule  and  routine.  The  mark  of  the  empiric 
is  that  he  is  helpless  in  the  face  of  uew  circumstances ; 
the  mark  of  the  scientitic  worker  is  that  he  has  power 
in  grappling  with  the  uew  and  the  untried ;  he  is  mas- 
ter of  principles  which  he  can  effectively  apply  under 
novel  conditions.  The  one  is  a  slave  of  the  past,  the 
other  is  a  director  of  the  future.  This  attachment  to 
routine,  this  subservience  to  empiric  formula,  always 
reacts  into  the  character  of  the  empiric;  he  becomes 
hour  by  hour  more  and  more  a  mere  routinist  aud  less 
and  less  an  artist.  Even  that  which  he  has  once  learned 
and  applied  with  some  interest  and  intelligence  tends 
to  become  more  and  more  mechanical,  and  its  a])})lica-- 
tion  more  and  more  an  unintelligent  and  unemotional 
procedure.  It  is  never  brightened  and  (piickened  by 
adaptation  to  new  ends.  The  machine  teacher,  like  the 
em])iric  in  every  profession,  thus  becomes  a  stupefying 
and  corrupting  inlluence  in  his  surroundings  ;  he  him- 


10 


THE   PSYCHOLOGY  OF  NUMBER. 


self  becomes  a  mere  tradesman,  and  makes  his  scliool  a 
mere  maebine  sbo[). 

3.  The  Tbird  Resource  is  authoritative  iihstruction 
in  methods  and  devices.  At  present,  tbe  real  opposi- 
tion is  not  ])et\veen  native  skill  and  experience  on  tbe 
one  side,  and  psycbological  metbods  on  tbe  otber ; 
it  is  ratber  between  devices  picked  up  no  one  knows 
how,  metbods  inherited  from  a  crude  ])ast,  or  else  in- 
vented, ad  hoc,  by  echicational  quackery — and  metbods 
which  can  be  rationally  justitied — devices  which  are 
tbe  natural  fruit  of  knowing  the  mind's  powers  and  tbe 
ways  in  which  it  works  and  grows  in  assimilating  its 
proper  nutriment.  The  mere  fact  that  there  are  co 
many  metbods  current,  and  constantly  pressed  upon  tbe 
teacher  as  tbe  acme  of  tbe  educational  experience  of 
the  past,  or  as  tbe  latest  and  best  discovery  in  peda- 
gogy, makes  an  absolute  demand  for  some  standard  by 
which  they  nuiy  be  tested.  Only  knowledge  of  tbe 
principles  upon  which  all  methods  are  based  can  free 
the  teacher  from  dependence  upon  the  educational  nos- 
trums which  are  recommended  like  patent  medicines, 
as  panaceas  for  all  educational  ills.  If  a  teacher  is  one 
fairly  initiated  into  tbe  real  workings  of  tbe  mind,  if  be 
realizes  its  iu)rmal  aims  and  metbods,  false  devices  and 
schemes  can  have  no  attraction  for  him  ;  be  will  not 
swallow  them  "as  silly  people  swallow  empirics'  pills"  ; 
he  will  reiect  them  as  if  bv  instinct.  All  new  suixires- 
tions,  new  metbods,  be  will  submit  to  tbe  infallible  test 
of  science  ;  and  those  which  will  further  bis  work  be 
can  ado})t  and  rationally  aj^ply,  seeing  clearly  their 
place  and  bearings,  and  the  conditions  under  which  they 
can  be  most  effectively  employed.     Tbe  difference  be- 


mmmm 


WHAT  PSYCIIOLOGY  CAN   DO  FOR  THE  TEACHER.   H 


a 


■m 


tween  being  overpowered  and  used  by  machinery  and 
being  al>le  to  use  the  machinery  is  ])recisely  the  dilier- 
ence  between  methods  externally  inculcated  and  meth- 
ods freely  adopted,  because  of  insight  into  the  psycho- 
logical principles  from  which  they  spring. 

Summing  u}),  we  may  say  that  the  teaclier  requires 
a  sound  knowledge  of  ethical  and  psychological  prin- 
ciples— first,  because  such  knowledge,  besides  its  indi- 
rect value  as  forming  logical  habits  of  mind,  is  necessary 
to  secure  the  full  use  of  native  skill  ;  secondly,  because 
it  is  necessary  in  order  to  attain  a  perfected  experience 
with  the  least  expenditure  of  time  and  energy  ;  and 
thirdly,  in  order  that  the  educator  may  not  be  at  the 
mercy  of  every  sort  of  doctrine  and  device,  but  may 
have  his  own  standard  by  which  to  test  the  many  meth- 
ods and  expedients  constantly  urged  upon  him,  select- 
ing those  which  stand  the  test  and  rejecting  those  which 
do  not,  no  matter  by  what  authority  or  influence  they 
may  be  supported. 

II.  We  may  now  consider  more  positively  how  psy- 
cliology  is  to  perform  this  function  of  developing  and 
directing  native  skill,  making  experience  rational  and 
hence  prolific  of  the  best  results,  and  providing  a  cri- 
terion for  ::.uij::<2:ested  devices. 

Education  has  two  nuiin  phases  which  are  never 
separated  from  each  other,  but  which  it  is  convenient 
to  distin<>;uish.  One  is  concerned  with  the  organization 
and  workings  of  the  school  as  part  of  an  organic  whole  ; 
the  other,  with  the  adaptation  of  this  school  structure 
to  the  individual  pupil.  This  difference  may  be  illustra- 
ted by  the  dift'erence  in  the  attitude  of  the  school  board 
or  minister  of  education    or  superintendent,  whether 


■)i| 


12 


THE  PSYCHOLOGY  OF   NUMBER. 


state,  county,  or  local,  to  the  school,  and  that  of  the 
individual  teacher  within  the  school.  The  former  (the 
administrators  of  an  organized  system)  are  concerned 
more  with  the  constitution  of  the  school  as  a  whole  ; 
their  survey  takes  in  a  wide  field,  extending  in  some 
cases  from  the  kindergarten  to  the  university  through- 
out an  entire  country,  in  other  cases  from  the  primary 
school  to  the  high  or  academic  school  in  a  given  town 
or  city.  Their  chief  husiness  is  with  the  organization 
and  management  of  the  school,  or  sj'stem  of  schools, 
upon  certain  general  principles.  What  shall  he  the  end 
and  means  of  the  entire  institution  'i  \Vhat  suhjects 
shall  he  studied  ?  At  what  stage  shall  they  he  intro- 
duced, and  in  what  sequence  shall  they  follow  one  an- 
other— that  is,  what  shall  be  the  ari-angement  of  the 
school  as  to  its  various  parts  in  time  (  Again,  what 
shall  be  the  correlation  of  studies  and  methods  at  every 
period  'i  Shall  they  be  taught  as  dilferent  subjects  ?  in 
departments  ?  or  shall  methods  be  sought  which  shall 
work  them  into  an  organic  whole?  All  this  lies,  in  a 
large  measure,  outside  the  purview  of  the  individual 
teacher  ;  once  within  the  institution  he  iinds  its  purpose, 
its  general  lines  of  work,  its  constitutional  structure,  as 
it  were,  fixed  for  him.  An  individual  mav  choose  to 
live  in  France,  or  (ireat  ]>ritain,  or  the  United  States, 
or  Canada;  but  afler  he  has  made  his  choice,  the  gen- 
eral ccndiiions  under  which  he  shall  exercise  his  citizen- 
Si»i-  i)  L  .VciMed  for  him.  So  it  is,  in  the  main,  with 
tb    ni'.i/I.^ial  tcachei'. 

ih\t  Aiv  "'.tizen  who  lives  within  a  given  system  of 
institutions  and  laws  hnds  himself  constantly  called  upon 
to  act.     He  must  adjust  his  interests  and  activities  to 


WHAT   PSYCIlOLO(iV  CAN   1)0  FOR  THE  TEACHER.   13 


a 


-\^, 


tliose  of  otliers  in  the  same  coiiiitrv.  There  is,  at  the 
same  time,  seope  for  purely  individual  seleetion  and 
application  of  means  to  ends,  for  unfettered  action  of 
strong  personality,  as  well  as  opportunity  and  stimulus 
for  the  free  ])layand  realization  of  individual  equipment 
and  acquisition.  The  hetter  the  constitution,  the  system 
which  he  can  not  directly  control,  the  wider  and  freer 
and  more  potent  will  he  this  sphere  of  individual  action. 
]Now,  the  individuiil  teacher  Unds  his  duties  within  the 
school  as  an  entire  institution  ;  he  has  to  adapt  this  or- 
ganism, the  suhjects  taught,  the  modes  of  discipline, 
etc,  to  the  individual  pupil.  Apart  from  this  personal 
adaptation  on  the  part  of  the  individual  teacher,  and 
the  personal  assimilation  on  the  part  of  the  individual 
pupil,  the  general  arrangement  of  the  school  is  purely 
meaningless  ;  it  has  its  object  and  its  justification  in 
this  individual  realm.  Geography,  arithmetic,  liter- 
ature, etc.,  may  be  provided  in  the  curriculum,  and 
their  order,  both  of  sequence  and  coexistence,  laid 
down  ;  but  this  is  all  dead  and  formal  until  it  comes 
to  the  intelligence  and  character  of  the  individual  pupil, 
and  the  individual  teacher  is  tlic  Diediiun  through  which 
it  co?nes. 

Now,  the  bearing  of  this  upon  the  point  in  hand 
is  that  psychology  and  ethics  have  to  subserve  these  two 
functions.  These  functions,  as  alreadv  intimated,  can 
not  be  separated  from  each  other  ;  they  are  simply  the 
general  and  the  individual  aspects  of  school  life  ;  but 
for  purposes  of  study,  it  is  convenient  and  even  impor- 
tant to  distinguish  them.  AVe  may  consider  psychology 
and  ethics  from  the  standpoint  of  the  light  they  throw 
upon  the  organization  of  the  school. as  a  whole — its  end, 


i  ' 


14 


THE   PSYCHOLOGY   OF   NTMBKR. 


its  chief  nietliods,  the  order  and  correlation  of  studies — 
and  we  niav  consider  them  from  the  stan<lii()int  of  tlie 
service  they  can  perform  for  the  individual  teacher  in 
qualifvinii;  him  to  use  tlie  prescribed  studies  and  meth- 
ods intelHijjentlv  and  efticiently,  th(,'  insio'ht  tliey  can 
ii-ive  liim  into  the  workino:s  of  the  individual  mind,  and 
the  relation  of  any  given  subject  to  that  mind. 

Next  to  positive  doctrinal  error  within  the  pedagogy 
itself,  it  may  be  said  that  the  chief  reason  why  so  much 
of  current  pedagogy  has  been  either  ])ractically  nseless 
or  even  practically  harmful  is  the  failure  to  di^tinguish 
these   two  functions   of   psycholoijy.      Considerations, 
])rinci|)Ies,  and  maxims  that  derive  their  meaning,  so 
fai*  as  they  have  any  meaning,  from  their  reference  to 
the  organization   of  the  wliole  institution,  have  been 
]n-esented  as  if  somehow  the  individual  teacher  might 
derive  from  them  specilic  information  and  direction  as 
to  how  to  teach  particular  subjects  to  particular  pupils  ; 
on  the  other  hand,  methods  that  have  their  value  (if 
any)  as  simple  suggestions  to  the  individual  teacher  as 
to  how  to  accomplish  temporary  ends  at  a  j)articular 
time  have  been  presented  as  if  they  were  eternal  and 
universal  laws  of  educational  polity.     As  a  result  the 
teacher  is  confused  ;  he  iinds  himself  expected  to  draw 
particular  practical  conclusions  from  very  vague  and 
theoretical  educational  maxims  (e.  g.,  ])roceed  from  the 
whole  to  the  ])art,  from  the  concrete  to  the  abstract),  or 
he  finds  himself  expected  to  adopt  as  rational  principles 
what  are  mere  temporary  expedients.     It  is,  indeed,  ad- 
visable that  the  teacher  should  understand,  and  even  be 
able  to  criticise,  the  general  pi'inciples  npon  which  the 
whole  educational  system  is  formed  and  administered. 


I 


wS»^!?f' 


WHAT  PSYCHOLOCiY  CAN  DO  FOR  THE  TEACHER.   15 


lie  is  not  like  a  private  soldier  in  an  army,  expected 
merely  to  obey,  or  like  a  cog  in  a  wheel,  expected 
merely  to  respond  to  and  transmit  external  enerp^y;  he 
must  be  an  intelligent  medium  of  action.  But  only 
confusion  can  result  from  trying  to  get  principles  or 
devices  to  do  what  they  are  not  intended  to  do — to  adapt 
them  to  purposes  for  which  they  have  no  fitness. 

In  other  words,  the  existing  evils  in  pedagogy,  the 
prevalence  of  merely  vague  principles  upon  one  side 
and  of  altogether  too  specific  and  detailed  methods 
(expedients)  npon  the  other,  are  really  due  to  failure 
to  ask  what  psychology  is  called  upon  to  do,  and  upon 
failure  to  present  it  in  such  a  form  as  will  give  it  un- 
doubted value  in  practical  applications. 

III.  This  brings  us  to  the  positive  question  :  In 
what  forms  can  psychology  best  do  the  work  -which  it 
ought  to  do  ? 

1.  The  PsychiGol  Functions  Mature  in  a  certain 
Order. — When  development  is  normal  the  appearance 
of  a  certain  impulse  or  instinct,  the  ripening  of  a  cer- 
tain interest,  always  prepares  the  way  for  another.  A 
i  child  spends  the  first  six  months  of  his  life  in  learning 
a  few  simple  adjustments  ;   his  instincts  to  reach,  to 

^   see,  to  sit  erect  assert  themselves,  and  are  worked  out. 
These  at  once  become  tools  for  further  activities  ;  the 

v^  child  has  now  to  use  these'  acquired  powers  as  means 

'^  for  further  accjuisitions.  Being  able,  in  a  rough  way, 
to  control  the  eye,  the  arm,  the  hand,  and  the  body  in 
certain  positions  in  relation  to  one  another,  he  now  in- 
spects, touches,  handles,  throws  what  comes  within  reach ; 

,.  and  thus  getting  a  certain  amount  of  physical  control, 

he  builds  up  for  himself  a  simple  world  of  objects, 
3 


10 


TIIH   PSYCHOLOGY   OF  NUMHEK. 


But  liis  instinctive  bodily  control  goes  on  asserting; 
itself  ;  he  continues  to  gain  in  ability  to  balance  liini- 
self,  to  co-ordinate,  and  thus  control,  the  movements  of 
Ids  body.  lie  learns  to  manage  the  body,  not  only  at 
rest,  but  also  in  motion — to  creep  and  to  walk.  Thus 
lie  gets  a  further  means  of  growth  ;  he  extends  his  ac- 
quaintance with  things,  daily  widening  his  little  world. 
lie  also,  through  moving  about,  goes  from  one  thing  to 
another — that  is,  makes  simple  and  crude  connections  of 
objects,  which  become  the  basis  of  subsequent  relating 
and  generalizing  activities.  This  carries  the  child  to 
the  aoje  of  twelve  or  fifteen  months.  Then  another  in- 
stinct,  already  in  occasional  operation,  ripens  and  takes 
the  lead — that  of  imitation.  In  other  words,  there  is 
now  the  attempt  to  adjust  the  activities  which  the  child 
has  already  mastered  to  the  activities  which  he  sees  exer- 
cised by  others.  He  now  endeavours  to  make  the  simple 
movements  of  hand,  of  vocal  organs,  etc.,  already  in  his 
possession  the  instruments  of  reproducing  what  his  eye 
and  his  ear  report  to  him  of  the  world  about  liim.  Thus 
he  learns  to  talk  and  to  repeat  many  of  the  simple  acts 
of  others.  This  period  lasts  (roughly)  till  about  the  thir- 
tieth month.  These  attainments,  in  turn,  become  the 
instruments  of  others.  The  child  has  now  control  of 
all  his  organs,  motor  and  sensory.  The  next  step,  there- 
fore, is  to  relate  these  activities  to  one  another  con- 
sciously, and  not  simply  unconsciously  as  he  has  hith- 
erto done.  When,  for  example,  he  sees  a  block,  he  now 
sees  in  it  the  possibility  of  a  whole  series  of  activities, 
of  throwing,  building  a  house,  etc.  The  head  of  a 
broken  doll  is  no  longer  to  him  the  mere  thing  directly 
before  his  senses.     It  symbolizes  "some  fragment  of 


WHAT  PSYCHOLOGY  CAN  DO  FOR  THE  TEACHER.  17 


m^ 


liis  (Iroain  of  human  life."  It  arouses  in  consciousness 
an  entire  group  of  related  actions  ;  the  child  strokes  it, 
talks  to  it,  sings  it  to  sleep,  treats  it,  in  a  word,  as  if 
it  were  the  perfect  doll.  AVhen  this  stage  is  reached, 
that  of  ability  to  see  in  a  partial  activity  or  in  a  single 
perce])tion,  a  whole  system  or  circuit  of  relevant  actions 
and  (jualities,  the  imagination  is  in  active  operation  ;  the 
period  of  symbolism,  of  recognition  of  meaning,  of  sig- 
nificance, has  dawned. 

But  the  same  general  process  continues.  Each  func- 
tion as  it  matures,  and  is  vigorously  exercised,  prepares 
the  way  for  a  more  comprehensive  and  a  deeper  con- 
scious activity.  All  education  consists  in  seizing  upon 
the  dawning  activity  and  in  presenting  the  material,  the 
conditions,  for  promoting  its  best  growth — in  making 
it  work  freely  and  fully  towards  its  proper  end.  Now, 
even  in  the  first  stages,  the  wise  foresight  and  direc- 
tion of  the  parent  accomplish  much,  far  more  indeed 
than  most  parents  are  ever  conscious  of  ;  yet  the  activ- 
ities at  this  stage  are  so  simple  and  so  imperious  that, 
given  any  chance  at  all,  they  work  themselves  out  in 
some  fashion  or  other.  But  wdien  the  stage  of  con- 
scious recognition  of  meaning,  of  conscious  direction 
of  action,  is  reached,  the  process  of  development  is 
much  more  complicated  ;  many  more  possibilities  are 
opened  to  the  parent  and  the  teacher,  and  so  the  de- 
mand for  proper  conditions  and  direction  becomes  in- 
definitely greater.  Unless  the  right  conditions  and 
direction  are  supplied,  the  activities  do  not  freely  ex- 
press themselves  ;  the  weaker  are  thwarted  and  die  out ; 
among  the  stronger  an  unhappy  conflict  wages  and  re- 
sults in  abnormal  grow^th  ;  some  one  impulse,  naturally 


ill 


!    '' 

I' 


t 

I 

f 


\'^ 


18 


THE   PSYCHOLOGY  OF  NUMHER. 


strons^er  than  others,  asserts  itself  out  of  all  proportion, 
and  the  person  ''runs  wild/'  becomes  wilful,  capricious, 
irresj)onsible  in  action,  and  unbalanced  and  irregular  in 
his  intellectual  operations. 

Onhj  I'notrleilye  of  the  order  and  connection  of  the 
staycH  in  the  development  (f  the pi^ychical functions  ca7i, 
negatively^  ijuard  atjaind  them  evils,  or,  j)o.siticeli/,  in- 
spire the  full  nuanihij  and  free,  yet  orderly  or  law- 
abidiny,  ed'ercise  if  tl^e i^sychical  -powers.  In  a  icord^ 
education  itself  is  precisely  the  icorl'  of  sujjjflyiny  the 
conditions  ichich  will  enahle  the  j)sychicalfnictions,  as 
they  successively  arise,  to  mature  and  j)ass  into  higher 
functions  in.  the  freest  and  fullest  manner,  and  this 
result  can  he  secured  only  hy  knowledge  of  the  j^rocess 
— that  is,  only  hy  a  knowledge  of  psychology. 

The  so-called  psychology,  or  ])edagogical  psychology, 
which  fails  to  give  this  insight,  evidertly  fails  of  its 
value  for  educational  purposes.  This  failure  is  apt  to 
occur  for  one  or  the  other  of  two  reasons  :  either  be- 
cause the  psychology  is  too  vague  and  general,  not  bear- 
ing directly  u])on  the  actual  evolution  of  psychical  life, 
or  because,  at  the  other  extreme,  it  gives  a  mass  of  crude, 
particular,  undigested  facts,  with  no  indicated  bearing 
or  interpretations : 

1.  The  psychology  based  upon  a  doctrine  of  ''  fac- 
ulties" of  the  soul  is  a  typical  representative  of  the 
first  sort,  and  educational  applications  based  upon  it  are 
necessarily  mechanical  and  formal ;  they  are  generally 
but  plausible  abstractions,  having  little  or  no  direct  ap- 
plication to  the  practical  work  of  the  classroom.  The 
mind  having  been  considered  as  split  up  into  a  number 
of  independent  powders,  pedagogy  is  reduced  to  a  set 


WHAT  PSYCHOLOGY  CAN  DO  FOR  THE  TEACHER.   19 


J", 

IS, 

in 


hi, 


of  precepts  about  the  "cultivation"  of  these  powers. 
These  precepts  are  useless,  in  the  first  })]ace,  because 
tlie  teacher  is  confronted  not  with  abstract  faculties, 
but  with  livin<^  individuals.  Even  when  the  psychology 
teaches  that  there  is  a  unitv  binding  tou'ether  the  va- 
rious  faculties,  and  that  they  are  not  really  separate, 
this  unity  is  presented  in  a  purely  external  way.  It  is 
not  shown  in  what  way  the  various  so-called  faculties 
are  the  expressions  of  one  and  the  same  fundamental 
process.  Jhit,  in  the  second  place,  this  "  faculty  "  psy- 
chology is  not  merely  negatively  useless  for  the  edu- 
cator, it  is  positively  false,  and  therefore  harmful  in  its 
effects.  The  psychical  reality  is  that  continuous  growth, 
that  unfolding  of  a  single  functional  principle  already 
referred  to.  While  perception,  memory,  imagination 
and  judgment  are  not  present  in  complete  form  from 
the  lirst,  one  psychical  activity  is  ])resent,  which,  as  it 
becomes  more  developed  and  con)plex,  manifests  itself 
in  these  processes  as  stages  of  its  growth.  AYliat  the 
educator  re(pnres,  therefore,  is  not  vague  information 
about  these  mental  powers  in  general,  but  a  clear  knowl- 
edge of  the  underlying  single  activity  and  of  the  con- 
ditions under  which  it  differentiates  into  these  powers. 

2.  The  value  for  educational  purposes  of  the  mere 
presentation  of  unrelated  facts,  of  anecdotes  of  child 
life,  or  even  of  particular  investigations  into  certain 
details,  may  be  greatly  exaggerated.  A  great  deal  of 
material  which,  even  if  intelligently  collected,  is  simply 
data  for  the  scientific  specialist,  is  often  presented  as  if 
educational  practice  could  be  guided  by  it.  Only  inter- 
jyreted  material,  that  which  reveals  general  principles 
or  suggests  the  lines  of  growth  to  which  the  educator 


I  I 


'U'i 


r,' 

Hi. 

1 

1 

1 

I  Ij 


\i 


i|!i 


11 


20 


THE   PSYCHOLOGY  OF   NUMBER. 


lias  to  adapt  himself,  can  be  of  nuicli  practical  avail ; 
and  the  interpreter  of  the  facts  of  the  child  mind 
must  begin  with  knowing  the  facts  of  the  adnlt  mind. 
E(pially  in  mental  evolution  as  in  pliysical,  nature 
makes  no  leaps.  "  The  child  is  father  of  the  man  "  is 
the  poetic  statement  of  a  psychologic  fact. 

3.  Every  Suhject  has  its  cncn  Psychological  Place 
and  Method. — Every  special  subject,  geography,  for  in- 
stance, represents  a  certain  grouping  of  facts,  classified 
on  the  basis  of  the  mind  'sattitude  towards  these  facts. 
In  the  thing  itself,  in  the  actual  world,  there  is  organic 
unity  ;  there  is  no  division  in  the  facts  of  geology, 
geography,  zoology,  and  botany.  These  facts  are 
not  externally  sorted  out  into  different  compartments. 
They  a?  e  all  bound  up  together ;  the  facts  are  many, 
but  the  thing  is  07ie.  It  is  simply  some  interest,  some 
urgent  need  of  man's  activities,  which  discritninates  the 
facts  and  unifies  them  under  diiferent  heads.  Unless 
the  fundamental  interest  and  jpnrpose  uihich  underlie 
this  classification  are  discovered  and  appealed  to^  the 
suhject  which  deals  with  it  can  not  he  presented  along 
the  lines  of  least  resistance  and  in  the  most  fruitfvl 
way.  This  discovery  is  the  work  of  psychology.  In 
geography,  for  example,  we  deal  with  certain  classes  of 
facts,  not  merel}^  in  themselves,  but  from  the  standpoint 
of  their  influence  in  the  development  and  modification 
of  human  activities.  A  mountain  range  or  a  river, 
treated  simply  as  mountain  range  or  river,  gives  us 
geology ;  treated  in  relation  to  the  distribution  of 
genera  of  plants  and  animals,  it  has  a  biological  inter- 
pretation ;  treated  as  furnishing  conditions  which  have 
entered  into  and  modified  human  activities — crrazinff. 


WHAT  PSYCHOLOGY  CAN  DO  FOR  THE  TEACHER.  21 


V 


transportation  of  commodities,  fixing  ]iolitical  bounda- 
ries, etc. — it  acquires  a  geographical  significance. 

In  other  words,  tlie  unity  of  geograj^hj  is  a  certain 
unity  of  human  action,  a  certain  human  interest.  Un- 
less, therefore,  geographical  data  are  presented  in  such 
a  way  as  to  appeal  to  this  interest,  the  method  of  teach- 
ing geography  is  uncertain,  vacillating,  confusing;  it 
throws  the  movement  of  the  child's  mind  into  lines  of 
great,  rather  than  of  least,  resistance,  and  leaves  him 
with  a  mass  of  disconnected  facts  and  a  feeling  of  un- 
reality in  presence  of  which  his  interest  dies  out.  All 
method  means  adaptation  of  means  to  a  certain  end  ;  if 
the  end  is  not  grasj)ed,  there  is  no  rational  principle  for 
the  selection  of  means ;  the  method  is  haphazard  and 
empirical — a  chance  selection  from  a  bundle  of  expe- 
dients. But  the  elaboration  of  this  interest,  the  discov- 
ery of  the  concrete  ways  in  which  the  mind  realizes  it, 
is  unquestional)ly  the  province  of  psychology.  There 
are  certain  definite  modes  in  which  the  mind  images  to 
itself  the  relation  of  environment  and  human  activity 
in  production  and  exchange ;  there  is  a  certain  order  of 
growth  in  this  imagery  ;  to  know  this  is  psychology ; 
and,  once  more,  to  know  this  is  to  be  able  to  direct  the 
teaching  of  geography  rationally  and  fruitfully,  and  to 
secure  the  best  results,  both  in  culture  and  discipline, 
that  can  be  had  from  the  study  of  the  subject. 

Applieat'uni  to  Arit/wietic— In  the  following  pnges 
an  attempt  has  l)een  made  to  present  the  psychology  of 
number  from  this  point  of  view.  Ts^umber  represents  a 
certain  interest,  a  certain  psychical  demand  ;  it  is  not  a 
bare  ]U'operty  of  facts,  but  is  a  certain  way  of  interpret- 
inir  and  arranmn":  them — a  certain  method  of  constru- 


■ih. 


t^ 


.i^ 


\  ■  1^. 


22 


THE  PSYCHOLOGY   OF  NUMBER. 


iiig  them.     AVhat  is  tlie  interest,  the  demand,  which 
gives  rise  to  the  psychical  activity  by  which  objects  are 
taken  as  numl)ered  or  measured  ?     And  how  does  this 
activity  develop  'i      In  so  far  as  we  can  answer  these 
(piestions  we  have  a  sure  guide  to  methods  of  instruc- 
tion in  dealing  with  number.     We  have  a  positive  basis 
for  testing  and  criticising  various  proposed   methods 
and  devices ;  we  have  only  to  ask  whether  they  are  true 
to  this  specilic  activity,  whether  they  build  upon  it  and 
further  it.     In  addition  to  this  we  have  a  standard  at 
our  disposal  for  setting  forth  correct  methods ;  we  have 
but  to  translate  the  theory  of  mental  activity  in  this 
direction  —  the   psychical   nature  of  number  and   the 
problem  of  its  origin — over  into  its  practical  meaning, 
knowing  the  nature  and  origin  of  number  and  numer- 
ical properties  as  psychological  facts,  the  teacher  knows 
how  the  mind  works  in  the  construction  of  number,  and 
is  prepared  to  help  the  child  to  think  number ;  is  pre- 
pared to  use  a  method,  helpful  to  the  normal  movement 
of  the  mind.     In  other  words,  rational  method  in  arith- 
metic must  be  based  on  the  psychology  of  number. 


CHAPTER  11. 


1    ■! 


THE    PSYCHICAL    NATURE    OF    NUMBER. 

Why  do  we  ask  with  respect  to  any  magnitude, 
"  how  many,"  "  how  much,"  and  set  about  counting 
and  measuring  till  we  can  say  "  so  many,"  "  so  much  "  ? 
Why  do  we  not  take  our  sense  experience  just  as  it 
comes  to  us,  making  no  attempt  to  give  it  these  exact 
quantitative  characteristics  ?  If  we  can  find  out  the 
psychological  reason,  the  mental  necessity,  which  in- 
duces us  to  put  our  experience  so  far  as  we  can  into 
terms  of  exact  measurement,  we  shall  have  a  principle 
which  will  guide  us  to  sound  conclusions  regarding  the 
nature  and  origin  of  number  and  its  rational  treatment 
as  a  school  study.  AYe  have  here  tacitly  assumed  that 
number  is  a  psychical  product,  and  has  a  psychical  rea- 
son for  its  origin.  Before  dealing  with  the  problem  of 
the  origin  of  number,  let  us  put  the  assumption  of  its 
psychical  nature  on  a  firmer  basis. 

Number  is  a  Rational  Process,  not  a  Sense  Fact. 
— The  mere  fact  that  there  is  a  multiplicity  of  things  in 
existence,  or  that  this  multi])licity  is  present  to  the  eye 
and  ear,  does  not  account  for  a  consciousness  of  num- 
ber. There  are  liundreds  of  leaves  on  the  tree  in  which 
the  bird  builds  its  nest,  but  it  does  not  follow  that  the 

bird  can  count. 

23 


If 


if 


I 


i 


i 


m 


24: 


THE  PSYCHOLOGY  OF  NUMBER. 


So  Iinndreds  of  noises  strike  the  ear,  and  countless 
objects  appeal  to  the  eye  of  a  cliild  a  few  weeks  old  ; 
but  he  is  not  conscious  of  the  noises  or  the  objects  as 
quantitative;  he  does  not  number  or  measure  them. 
More  than  this,  sense  facts  nuiv  be  even  attended  to 
without  giving  the  idea  of  number.  To  put,  say,  five 
objects  before  an  older  child,  to  call  his  mind  away 
from  all  other  things  and  get  his  attention  fixed  upon 
these  objects,  is  not  to  give  him  the  idea  of  the  number 
five.  ^'^ umber  is  not  a  property  of  the  objects  which 
can  be  realized  through  the  mere  use  of  the  senses,  or 
impressed  upon  the  mind  by  so-called  external  energies 
or  attributes.  Objects  (and  measured  things)  aid  the 
mind  in  its  work  of  constructing  numerical  ideas,  but 
the  objects  are  not  number.  Kor  does  the  bare  percep- 
tion of  them  constitute  number.  A  child,  or  an  adult, 
may  perceive  a  collection  of  balls  or  cubes,  or  dots  on 
paper,  or  a  bunch  of  bananas,  or  a  pile  of  silver  coins, 
without  an  idea  of  their  number  ;  there  may  be  clear 
and  adequate  perce})ts  of  the  things  quite  unaccom- 
panied by  definite  numerical  concepts.  No  such  con- 
cepts, no  clearly  defined  numerical  ideas,  can  enter  into 
consciousness  till  the  mind  orders  the  objects — that  is, 
compares  and  relates  them  in  a  certain  way. 

Factoks  of  thk  Intellectual  Process. — In  the  sim- 
j)le  recognition,  for  example,  of  three  things  as  three  the 
foHowing  intellectual  operations  are  involved  :  T/te  rec- 
iHjn'dion  of  the  three  ohjeeis  as  forming  one  connected 
whole  or  (jrouj) — that  is,  there  must  be  a  recognition  of 
the  three  things  as  individuals,  and  of  the  one,  the  unity, 
the  whole,  made  up  of  the  three  things.  If  one  of  the 
objects  is  a  piece  of  candy,  and  the  other  two  are  dots 


THE  PSYCHICAL  NATURE  OF  NUiMBER. 


25 


on  paper,  the  candy  may  so  absorb  attention  that  the 
two  dots  do  not  present  themselves  in  consciousness  at 
all.  This  is  undoubtedly  one  reason  why  the  mathe- 
matical attainments  of  savages  are  so  meagre  ;  they  are 
so  given  up  to  one  absorbing  thing — which  is  to  them 
what  the  candy  is  to  the  child — that  the  rest  of  tlie  uni- 
verse, however  much  it  may  affect  their  senses,  does  not 
become  an  object  of  attention. 

Or,  again,  tlie  child  may  be  conscious  of  the  dots  as 
wxU  as  of  the  candy,  and  yet  not  be  able  to  recognise 
that  these  various  objects  are  connected  or  make  one 
whole.  The  qualitative  unlikeness  of  the  objects  may 
be  so  great  as  to  make  it  difficult  or  even  impossible  for 
the  child's  mind  to  relate  them,  to  view  them  all  from 
a  common  standpoint  as  forming  one  group.  The  candy 
is  one  thing  and  the  dots  are  another  and  entirely  dif- 
ferent thing.  Here,  again,  rational  counting  is  out  of 
the  question. 

Nor,  finally,  is  it  to  be  concluded  that  from  the  mere 
presentation  of  three  like  objects  the  idea  of  three  will 
be  secui-ed.  There  must  be  enough  qualitative  unlike- 
ness— if  only  of  position  in  space  or  sequence  in  time — 
to  mark  off  the  individual  objects,  to  keep  them  from 
fusing  or  running  into  one  vague  whole.  Part  of  the 
difficulty  of  performing  the  abstraction  which  is  re- 
quired to  get  the  idea  of  number  is,  accordingly,  that 
this  abstraction  is  c(>inph\i\  involving  tJTO_f,fi(^tni\^ ;  the 
dilference  which  makes  the  individuality  of  each  obiect 
must  be  noted,  and  yet  the  different  individuals  must 
1)0  grasped  as  one  whole — a  sum.  It  recpdres,  then, 
considerable  power  of  intellectual  abstraction  even  to 
count  three.     Unlike  objects,  in  spite  of  differences  in 


1 1 


m 


I  --h 


THE   PSYCHOLOGY   OF  NUMBER. 


quality,  must  be  recognised  as  forming  one  grcnp ; 
while  a  group  of  like  objects,  in  spite  of  their  simi- 
larities in  quality,  is  to  be  recognised  as  made  up  of 
separate  things.  Three  dilferently  coloured  cubes,  for 
example,  must  be  apprehended  as  one  group,  while  a 
group  of  three  cid)es  exactly  alike  must  be  apprehended 
as  three  individurls.  \\  other  words,  the  objects  count- 
ed, whatever  be  their  physical  resemblances  or  differ- 
ences, are  numerically  alike  in  this  :  they  are  parts  of 
one  u'Jiole — they  are  rniif'^  constituting  a  defined  miity. 
The  delight  whfci.  h  vl.ild  four  or  five  years  old 
often  manifests  in  the  apjc,\;ntly  mechanical  operation 
of  counting  chairs,  books,  ^'a^e-marks,  playthings,  or 
even  in  simply  saying  ove  l"l'e  •  i^jics  of  number  sym- 
bols is  really  delight  in  his  novvl}  if  piired  or  rapidly 
growing  power  of  abstraction  and  generalization.  There 
is  abstraction  because  the  child  now  knows,  in  a  definite, 
objective  way,  that  one  chair,  although  a  different  chair 
from  every  other,  is,  nevertheless,  in  some  particular 
identical  with  every  other — it  is  a  chair,  lie  is  able  to 
neglect  all  that  sensuous  qualitative  difference  which 
previously  so  claimed  his  attention  as  to  prevent  his 
conscious  or  objective  recognition  of  the  common  qual- 
ity or  use  through  which  the  things  may  be  classed  as 
one  whole.  Xow,  abiHty  to  neglect  certain  features  of 
things  in  view  of  another  considered  more  important, 
is  of  course  of  the  essence  of  abstraction  in  its  hii>:liest 
as  well  as  in  this  rudimentary  form.  Generalization,  on 
the  other  hand,  is  simj^ly  the  obverse  of  abstraction  ; 
they  are  correlative  phases  of  one  activity.  In  leaving 
out  of  account  the  qualities  now  seen  to  be  unimportant 
tu  the  end  in  view,  though  sensuously  they  may  be  very 


THE    PSYCHICAL  NATURE  OF  NUMBER. 


27 


prominent  and  attractive,  tlie  mind  grasps  in  one  wliole 
the  objects  that  liave  a  coimHon  quality  or  use,  tliougli 
the  objects  are  decidedly  nidike  as  regards  other  quali- 
ties or  uses.  If  from  a  collection  of  ol)jects  of  different 
colours  a  cliild  is  required  to  select  all  the  red  ones,  he 
not  only  neglects  all  that  are  not  red  ;  he  neglects  also 
all  the  other  qualities — shape,  size,  material,  etc. — of  the 
red  objects  themselves ;  and  when  this  abstraction  is 
completed,  there  is  the  conception  of  the  group  of  red 
things  as  the  result  of  the  other  side  of  the  mental 
process — viz.,  generalization. 

The  manifestation  of  the  conscious  tendency  in  a 
cliild  to  count  coincides,  then,  with  the  awakening  in 
his  mind  of  conscious  power  to  abstract  and  generalize. 
This  power  can  show  itself  only  when  there  is  ability  to 
resist  the  immediate  solicitations  of  colour,  sound,  etc., 
ability  to  hold  the  mind  from  being  absorbed  in  the 
delight  of  mere  seeing,  hearing,  handling ;  and  this 
means  power  of  abstraction.  But  this  very  power  to 
resist  the  stimulus  of  some  sense  qualities  and  to  attend 
to  others  means  also  the  power  to  group  the  different 
objects  together  on  the  basis  of  some  principle  not  di- 
rectly apprehended  by  the  senses — some  use  or  function 
which  all  the  different  objects  have — and  this  is,  again, 
generalization. 

Discrimination  and  Kelation. — This  power  to  form 
a  whole  out  of  different  objects  may  be  studied  in  some- 
what more  detail.  It  includes  the  two  correlative  pow- 
ers of  discrimination  and  relation. 

1.  Discrimination. — As  adults  we  are  constantly 
deceiving  ourselves  in  regard  to  the  nature  and  genesis 
of  our  mental  experiences.     Because  an  object  presents 


i:; 


M! 


i 

,! 

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i 

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1 

28 


THE   PSYCHOLOGY   OF   NUMIiKU. 


a  certain  quality  directly  to  us,  we  are  apt  to  assume 
that  the  (quality  is  inherent  in  the  object  itself,  and  is 
])resented  to  everybody  (juite  apart  from  any  intellectual 
operation.  AV^e  forget  that  the  objects  now  have  certain 
qualities  for  us  KiiKply  because  of  analyses  j^/v^ij^-* ?/*/// 
l^erformed.  We  see  in  an  object  just  what  we  ha\e 
learned  to  see  in  it.  The  contents  of  the  concept  re- 
sultino^  from  an  elaborate  process  of  analysis-synthesis 
arc  at  last  given  in  the  percept.  An  expert  geometri- 
cian's percept  of  a  triangle  is  quite  a  different  thing 
from  that  of  a  mere  tyro  in  cjeometrv.  A  man  may 
become  such  a  chemist  as  never  to  see  water  without 
being  conscious  that  it  is  composed  of  oxygen  and  hydro- 
gen ;  or  such  a  botanist  that  a  passing  glance  at  a  liower 
instantly  recalls  the  name  orchid,  or  ranunculus,  and  all 
the  differential  qualities  which  belong  to  this  class  of 
plant  life.  In  like  manner  all  of  us  have  become  suffi- 
ciently familiar  with  numerical  ideas  to  know  at  a  glance 
that  a  tree  has  a  great  many  lefives,  a  chair  a  certain 
number  of  parts,  a  cube  a  definite  numl)er  of  faces. 
Althouijh  this  knowledo-e  is  now  direct  and  "intuitive," 
it  is  the  result  of  past  discriminations.  AYe  may  be 
perfectly  sure  that  they  are  not  "intuitions"  to  the 
child  ;  to  him  the  tree,  the  house,  the  cube,  the  black- 
board, the  group  of  six  objects,  is  one  undefined  whole, 
not  a  whole  of  parts.  The  recognition  of  separate  or 
distinct  parts  always  implies  an  act  of  analysis  or 
discrimination  definitely  performed  at  some  period ^ 
and  such  definite  analysis  has  always  been  preceded 
by  a  vague  synthesis — that  is,  the  idea  of  a  whole  of 
as  yet  undistinguished  parts. 

There  is  perhaps  no  point  at  which  the  teacher  is 


^ 


THE   PSYCIIK'Ah   NATURE  OF  NUMBER. 


'20 


more  likely  to  c;o  astray  than  in  assuming  that  objects 
liave  for  a  child  the  deiiniteness  or  concreteness  of 
qualities  which  they  have  for  us.  In  the  application 
of  the  pedagogical  maxim  "  from  the  concrete  to  the 
abstract,"  he  is  very  apt  to  overlook  the  necessity  of 
making  sure  that  the  "concrete"  is  really  present  to 
the  child's  mind.  lie  too  easily  assumes  as  already  ex- 
isting in  tlie  consciousness  of  the  learner  what  can 
really  exist  only  as  the  product  of  the  mind's  own 
activity  in  the  process  of  deiinition — of  discriminating 
and  relating.  It  is  a  grave  error  to  suppose  that  a  ti-i- 
angle,  a  circle,  a  written  word,  a  collection  of  five  ob- 
jects, are  concrete  wholes,  that  is,  definitely  grasped 
mental  wholes  to  the  child,  simply  because  there  are 
certain  physical  wholes  present  to  his  senses.  Definite 
ideas  are  thus  assumed  as  the  basis  of  later  work  when 
there  is  absolutely  nothing  corresponding  to  them  in 
the  child's  mind,  in  wliich,  indeed,  there  is  only  a  pano- 
rama of  vague  shifting  imagery,  with  a  penumbra  of 
all  sorts  of  irrelevant  emotions  and  ideas.  Thus,  this 
noted  maxim,  when  translated  to  inean  co7icTete  tJiings 
hi  fore  the  senses^  thei^efore  concrete  hnoudedge  in  the 
mind.,  becomes  really  a  mischievous  fallacy. 

Or,  again,  the  teacher,  mislead  by  the  formula — first, 
the  isolated  definite  particular ;  second,  the  interconnec- 
tion ;  third,  the  organic  whole — introduces  distinction 
and  definition  where  normally  the  child  should  deal 
only  with  wholes  in  vague  outline  ;  and  thus  substitutes 
for  the  poetic  and  spontaneous  character  of  mental  ac- 
tion a  forced  mechanical  analysis  all  out  of  harmoTiy 
with  his  existiiig  stage  of  development.  Of  this  we 
have  an  example  in  the  prevailing  methods  of  primary 


t ' 


1.1 


"\ 


h  I 


i '} 


30 


THE   PSYCHOLOGY   OF   NUMBER. 


M 


,y  number  teaching.  The  child  is  from  the  beginning 
drilled  in  the  "analysis"  of  numbers  till  he  knows  or 
is  supposed  to  know  '•  all  that  can  be  done  with  num- 
bers." It  appears  to  be  forgotten  tliat  he  may  and 
should  perform  many  0})e rations  and  reach  definite  re- 
sults by  implicitly  'iising  the  ideas  they  involve  long 
before  these  ideas  can  be  explicitly  developed  in  con- 
sciousness. If  facts  are  presented  in  their  proper  con- 
nection as  stimulating  and  directing  the  primary  mental 
activities,  the  child  is  slowly  but  surely  feeling  his  way 
towards  a  conscious  recognition  of  the  nature  of  the 
process.  This  unconscious  growth  towards  a  reflective 
grasp  of  number  relations  is  seriously  retarded  by  un- 
timely analysis — untimely  because  it  appeals  to  a  power 
of  reflection  wdiicli  is  as  yet  undeveloped. 

It  is  obvious  that  these  two  errors  are  logically  op- 
posed to  each  other.  One  overlooks  the  need  of  the 
process  of  discrimination,  of  careful  analysis ;  the  other 
does  nothing  but  analyze  and  deflne.  But  while  log- 
ically opposed  to  each  other  they  are  often  practically 
combined.  They  both  arise  from  one  fundamental 
error — the  failure  to  grasp  clearly  the  place  which  dis- 
crimination occupies  as  the  transitional  step  in  the 
change  of  a  vague  whole  into  a  coherent  whole.  In 
the  ordinary  methods  of  teaching  number,  for  example, 
both  mistakes  are  found  in  combination.  There  is  no 
attention,  or  too  little  attention,  paid  to  the  essential 
process  of  discrimination  when  it  is  taken  for  granted 
that  definite  ideas  of  number  will  be  formed  from  the 
hearing  and  memorizing  of  numerical  tables,  or  even 
from  the  perception  of  certain  objects  apart  f rani  the 
child's  own  activity  in  conceiving  a  whole  of  pai'ts  and 


\ 


THE   PSYCHICAL   NATURE  OF   NUMBER. 


31 


W 
V 


relating  j)arts  in  a  definite  whole.  On  the  other  htuul, 
there  is  altogether  too  much  definition,  definition  car- 
ried to  the  point  of  isolation,  when,  in  number  teach- 
ing, a  start  is  made  with  one  thing — endless  changes 
being  rung  with  single  objects  in  order  "  to  develop 
the  number  one  " — then  another  object  is  introduced, 
then  another,  and  so  on.  Here  the  preliminary  ac- 
tivity that  resolves  a  whole  into  parts  is  omitted,  as 
well  as  the  connecting  link  that  makes  a  lohole  of  all 
the  parts. 

2.  Relation  or  Rational  Cou7itln(/.--T\i\s  involves 
the  putting  of  units  (parts)  in  a  certain  ordered  relation 
to  one  another,  as  well  as  marking  them  off  or  dis- 
criminating them.  If,  when  the  child  discriminates 
one  thing  from  another,  he  loses  sight  of  the  identity, 
the  link  which  connects  them,  he  gains  no  idea  of  a 
group,  and  hence  there  is  no  counting.  There  is,  to 
him,  simply  a  lot  of  unrelated  things.  When  we  reach 
''  two  "  in  counting,  we  must  still  keep  m  7nind  ^'one^' ; 
if  we  do  not  we  have  not  "  two,"  but  merely  another 
one.  Two  things  may  be  before  us,  and  the  word 
*'  two  "  may  be  uttered  but  the  concept  two  is  absent. 
The  concept  two  involves  the  act  of  putting  together 
and  holding  together  the  two  discriminated  ones.  It  is 
this  tension  between  opposites  which  is  largely  the  basis 
of  the  childish  delight  in  counting.  Number  is  a  con- 
tinued paradox,  a  continued  reconciliation  of  contradic- 
tions. If  two  things  are  simply  fused  in  each  other, 
forming  a  sort  of  vague  07ieness,  or  if  they  are  6im})ly 
kept  apart  from  each  other,  there  is  no  counting,  no 
"  two."  It  is  the  correlative  differentiation  and  identi- 
fication, the  holding  apart  and  at  the  same  time  bringing 
4 


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32 


TI1I<:   PSVCIlOLOCiY   OF   NUMBiai. 


together,  wliicli  imparts  to  the  operation  of  coiniting  its 
faHciiuition.  This  activity  is  simply  the  normal  exercise 
of  what  are  always  the  fundamental  rational  functions  ; 
and  thus  it  gives  to  the  child  the  same  sense  of  power, 
of  ease  and  mastery  in  mental  movement,  tliat  an  adult 
may  realize  from  some  magnificent  generalization 
through  which  a  vast,  disorderly  lield  of  experience  is 
reduced  to  unity  and  system.  In  the  simple  one,  two, 
three,  four  of  the  child,  as  he  counts  the  familiar  ob- 
jects around  liim,  there  is  presented  the  form  of  the 
highest  operations  of  discrimination  and  identification. 

Educational  Summary. — 21\e  idea  of  'niimher  is 
not  impressed  upon  the  mind  hy  objects  even  when  these 
are pt'(^sented  under  the  most  feivovredde  circumstances. 
N'und)er  is  a  ji't'oduct  of  the  tcay  in  ichlch  the  mind 
deals  imtli  objects  in  the  operation  of  maMng  a  vagne 
whole  defnite.  This  operation  involves  («)  discrimi- 
7iatio?i  or  the  recognition  of  the  objects  as  distinct  indi- 
viduals (units) ;  (h)  generalisation,  this  latter  activity 
involving  two  subprocesses  ;  (J)  abstraction,  the  neg- 
lecting of  all  characteristic  qualities  save  just  enough  to 
limit  each  object  as  (me  ;  and  (2)  grovping,  the  gather- 
ing together  the  like  objects  (units)  into  a  whole  or 
class,  the  sum.     Hence  : 

1.  Kumber  can  not  be  taught  by  the  mere  presen- 
tation of  things,  but  only  by  snch  presentation  as  will 
stimulate  and  aid  the  mental  movement  of  discriminat- 
ing, abstracting,  and  grouping  which  leads  to  definite 
numerical  ideas. 

2.  In  this  process  there  must  be  sufficient  qualitative 
difference  among  the  objects  used  to  facilitate  the  recog- 
nition of  individuals  as  distinct,  but  not  enough  to  resist 


Till-:   PSYCHICAL   NATURE  OF   NUMIJHU. 


'.^ 


tlio  power  of  <>^roupini^  nil  the  iiidividiuils,  of  gnispiiiij 
them  us  j^Jirts  of  one  whole  or  sum. 

The  applieution  of  this  principle  will  depend  largely 
upon  circumstanees  (sensory  aptitudes,  ete.)  and  the  tact 
of  the  teacher.  In  some  cases  it  may  he  well  at  the 
outset  to  use  differently  coloured  cubes,  the  different 
colours  serving  to  individualize  each  ol)ject  or  group  of 
objects  as  a  unit,  while  the  common  cubical  quality 
facilitates  relation.  In  other  cases  the  diiferen.'e  in 
colour  might  divert  attention  from  the  relating  process, 
and  hinder  the  grasping  of  the  different  units  as  one 
Slim  I  the  mere  difference  of  position  in  space  would 
be  enough  for  the  necessary  discrimination. 

3.  In  any  case  the  aim  must  be  to  enable  the  pupil  to 
get  along  with  the  minimum  of  actual  sense  dift'erence, 
and  thus  further  the  power  of  mathematical  abstrac- 
tion and  relation.  For  discrimination  must  operate  just 
enough  for  the  recognition  of  the  individuality  or  sin- 
gleness of  eacli  object  or  part,  and  no  further.  The 
end  is  the  facile  recognition  of  groups  as  (jrouj)s^  the 
individuals,  the  single,  component  parts  being  consid- 
ered not  for  their  own  sake,  but  simply  as  giving  defi- 
nite value  to  the  group.  That  is  to  say,  the  recognition, 
for  instance,  of  three,  or  four,  or  five,  must  be  as  nearly 
as  possible  an  intuition ;  a  perception  of  the  parts  in 
the  whole  or  a  whole  of  parts,  and  not  a  conscious  recog- 
nition of  each  part  by  itself,  and  then  a  conscious  unit- 
ing it  to  other  parts  separately  recognised. 

4.  It  is  clear  that  to  promote  the  natural  action  of 
the  mind  in  constructing  number,  the  starting  point 
should  be  not  a  single  thing  or  an  unmeasured  whole, 
but  a  group  of  things  or  a  measured  whole.     Attention 


.  I 


11 


I. 


34  THE  PSYCHOLOGY  OP  KUiMBER. 

fixed  upon  a  single  nnmeasured  object  will  discriminate 
and  unify  the  qualities  which  make  the  thing  a  qualita- 
tive whole,  but  can  not  discriminate  and  relate  the  parts 
which  make  the  thing  a  definite  quantitative  whole.  It 
is  equally  clear  that  with  groups  of  things  the  move- 
ment in  numerical  abstracting  and  relating  may  be 
greatly  assisted  by  the  arrangement  of  the  things  in 
analytical  forms,  as  is  the  case,  e.  g.,  with  the  points 
on  dominoes. 


(I  t 


CHAPTER  III. 

THE  ORIGIN  OF  NUMBER  :  DEPENDENCE  OF  NUMBER  ON 
MEASUREMENT,  AND  OF  MEASUREMENT  ON  ADJUSTMENT 
OF  ACTIVITY. 

Admitting,  then,  the  psychical  iiaturo  of  number, 
we  are  now  prepared  to  deal  with  its  psychological 
origin.  It  does  not  arise,  as  we  have  seen,  from  mere 
sense  perception,  but  from  certain  rational  processes  in 
construing,  in  defining  and  relating  the  material  of 
sense  perception.  But  we  are  not  to  suppose  that  these 
processes — numerical  abstraction  and  generalization — 
account  for  themselves.  They  give  rise  to  number,  but 
tiierc  IS  some  reason  why  we  perform  them.  This  rea- 
son we  must  now  discover,  for  it  lies  at  the  root  of  the 
problem  of  the  origin  of  numher. 

The  Idea  of  Limit. — If  every  human  being  could 
use  at  his  pleasure  all  the  land  he  wanted,  it  is  probable 
that  no  one  would  ever  measure  land  with  mathematical 
exactness.  There  might  be,  of  course — Crusoe-like — 
a  crude  estimate  of  the  quantity  required  for  a  given 
purpose  ;  but  there  would  be  no  definite  numerical  val- 
uation in  acres,  rods,  yards,  feet.  There  would  be  no 
need  for  such  accuracy.  If  food  could  be  had  without 
trouble  or  care,  and  in  sufficiency  for  everybody,  we 
should  never  put  our  berries  in  (puirt  measures,  count 

85 


;;  \\ 


l!] 


r'-  ; 


i 


36 


THE  PSYCHOLOGY  OP  NUxMBKR. 


off  eggs  and  oranges  by  the  dozen,  and  weigh  out  flonr 
by  the  pound.  If  everything  tliat  ministers  to  liunian 
wants  could  be  had  by  everybody  just  when  wanted,  we 
should  never  have  to  concern  ourselves  about  quantity. 
If  everything  with  which  human  activity  is  in  any  way 
concerned  were  unlimited,  thei-e  would  of  course  be  no 
need  to  inquire  respecting  anything  whatever :  What 
are  its  limits  ?  How  much  is  there  of  it?  Even  if  a 
thing  were  not  actually  unlimited,  if  there  were  always 
enough  of  it  to  be  had  with  little  or  no  expenditure  of 
energy,  it  would  be  practicalJij  unlimited,  and  hence 
would  never  be  measured.  It  is  because  we  have  to 
-  put  forth  effort,  because  we  have  to  take  trwible  to  get 
things,  that  tliey  are  limited  for  us,  and  that  it  becomes 
worth  while  to  determine  their  limits,  to  iind  out  the 
quantity  of  anything  with  which  human  energy  has  to  do. 
Limit,  in  other  words,  is  the  primary  idea  in  all 
quantity ;  and  the  idea  of  limit  arises  because  of  some 
resistance  met  in  the  exercise  of  our  activity. 

Economy  of  Energy. — Because  we  have  to  put  forth 
effort,  because  we  are  confronted  by  obstacles,  our  enei-gy 
is  limited.  It  therefore  l)ecomes  necessary  to  economize 
our  energy — that  is  to  say,  to  dispose  of  it  or  disti'ibute 
it  in  such  ways  as  will  accomplish  the  best  j)ossible  re- 
sults. This  economy  does  not  mean  a  hoarding  up  or 
withholding  of  energy,  but  rather  (jiving  it  ont  in  the 
most  effective  way^  husbanding  "  oiT-r  means  so  well 
they  shall  go  far."  If  we  put  forth  more  energy  than 
is  needed  to  effect  a  certain  purpose,  and  ecpially  if  we 
])ut  forth  less  than  is  needed,  there  is  waste  ;  we  fail  to 
make  the  most  of  the  resources  at  our  disposal.  We 
carry  out  our  plans  most  successfully,  and  perform  the 


THE  ORIGIN  OF  NUMBER. 


37 


V 


hardest  tasks  ^vitll  the  least  waste  of  power  when  we 
accurately  adjust  our  energies  to  the  thing  required. 
Because  of  the  limitation  of  human  energy  all  activity 
is  a  halancing  of  energy  over  against  the  thing  to  be 
done,  and  is  most  fruitful  of  results  when  the  balancing 
is  most  accurate.  If  the  arrow  of  the  savage  is  too 
heavy  for  his  bow,  or  if  it  is  too  light  to  pierce  the  skin 
of  the  deer,  there  is  in  both  cases  a  waste  of  energy.  If 
the  bow  is  so  thick  and  clumsy  that  all  his  strength  is 
required  to  bend  it,  or  so  slight  or  uneven  that  too 
little  momentum  is  given  to  the  arrow,  there  is  but  a 
barren  show  of  action,  and  the  savage  has  his  labour  for 
his  pains.  Bow  and  arrow  must  be  accurately  adjusted 
to  each  other  in  size,  form,  and  weight ;  and  both  have 
to  be  equated  (as  the  mathematician  would  say)  or  bal- 
anced to  the  end  in  view — the  killing  of  the  game.  This 
involves  the  process  of  measurement,  and  its  result  is 
more  or  less  definite  numerical  values. 

Means  and  End :  Valuation. — The  same  ])rinciple 
may  be  otherwise  stated  in  terms  of  the  relation  exist- 
ing between  means  and  end.  If  all  our  aims  were 
reached  at  the  moment  of  forming  them,  without  any 
delay,  postponement,  or  countervening  occurrences — if 
to  realize  an  end  we  had  only  to  conceive  it — the  neces- 
sity for  measurement  would  not  exist,  and  there  would 
be  no  such  thing  as  number  in  the  strictly  mathematical 
sense.  But  the  check  to  our  activity,  the  limitation  of 
energy,  defers  the  satisfaction  of  our  needs.  The  end 
to  be  realized  is  remote  and  complex,  and  in  using 
adecjuate  means,  distance  in  space,  remoteness  in  time, 
<piantity  of  some  sort  has  to  be  taken  into  account, 
and  this  means  accurate  measurement. 


1 1 


\  f\ 


1  i  *^ 


38 


THE  PSYCHOLOGY  OF  NUMBER. 


In  working  out  a  certain  purpose,  for  example,  one 
of  a  series  of  means  is  a  journey  to  be  undertaken  ;  it 
is  of  a  certain  length  ;  it  is  to  be  completed  in  a  given 
time,  and  within  a  certain  maximum  of  expense,  etc. ; 
and  this  involves  careful  calculation — measurement  and 
immerical  ideas.  In  brief,  it  may  be  said  that  qxiantity 
enters  into  all  the  activities  of  life,  that  the  limitation 
of  energy  demands  its  economical  nse — that  is,  the  pre- 
cise adjustment  of  means  to  end — and  that  such  nse, 
such  careful  adjustment  of  activity,  depends  upon  exact 
measurement  of  quantity. 

The  child  and  the  savage  have  very  imperfect  ideas 
of  number,  because  they  are  taken  up  with  the  things 
of  the  present  moment.  There  is  no  imperative  de- 
mand for  the  economical  adjustment  of  means  to  end  ; 
living  only  in  and  for  the  present,  they  have  no  plans 
and  no  distant  end  recpiiring  such  an  adjustment.  They 
do  not  ask  how  the  present  is  to  be  made  of  nse  in  at- 
taining some  future  or  permanent  good.  But  as  soon 
as  the  child  or  the  savage  has  to  arrange  his  acts  in  a 
certain  order,  to  prescribe  for  himself  a  certain  course  of 
conduct  so  as  to  accomplish  something  remote,  then  the 
idea  of  qnantity  begins  to  exist.  When  a  savage  is  aim- 
lessly playing  with  a  stick,  he  does  not  connect  it  with 
any  desired  end,  and  accordingly  does  not  reflect  upon 
its  quantitative  value.  But  if  he  wants  to  shape  it  into 
an  arrow,  then  the  measuring  (the  quantifying)  imme- 
diately begins ;  he  examines  several  sticks  ;  he  thinks 
this  one  too  big,  that  too  small ;  this  too  brittle,  that  too 
elastic;  this  one  of  the  right  size,  weight,  and  elasticity 
— all  of  which  are  simple  quantitative  ideas. 

Thus  it  is  also  in  the  case  of  a  child  playing  with 


THE  ORIGIN   OF   NUMBER. 


39 


stones ;  so  long  as  he  is  contented  with  them  just  as 
they  are,  not  thinking  of  them  as  means  towards  a  definite 
end,  ideas  as  to  their  weight,  or  size,  or  immber,  do  not 
enter  into  his  mind.  But  if  he  decides  to  throw  at  a 
mark,  a  rude  measurement  or  vahiation  begins  ;  this  is 
too  lieavy  or  too  large,  that  too  light  or  too  small,  etc. 
Or  if  he  wishes  to  build  a  house,  or  to  form  an  inclosure 
with  them,  he  begins  at  once  to  note  size  and  shape,  and 
perhaps  to  form  a  vague  notion  of  the  number  required 
for  his  ideal  house  or  inclosure.  Having  only  the  vaguest 
ideas  of  quantity  and  number,  he  can  not  accurately 
compare  means  with  end,  and  his  first  efforts  at  building 
wull  be  purely  tentative.  His  ideal  "playhouse"  or 
sheepfold,  or  garden,  is  too  large  for  his  means ;  he 
has  not  stones  enough  to  complete  the  house  or  to  "go 
round  "  the  inclosure ;  he  must  build  on  a  smaller  scale. 
Or  the  structures  are  completed  without  exhausting  the 
materials,  and  with  the  remaining  stones  he  puts  to- 
gether another  piece  of  work,  perhaps  an  addition  to 
house  or  garden.  In  all  this  there  is  more  careful  com- 
parison of  quantities,  a  better  adjustment  of  means  to 
end,  closer  measurement,  and  some  approach  to  definite 
numerical  ideas. 

Again,  the  savage  has  to  take  a  journey  to  find  his 
game.  The  remoteness  of  the  hunting  ground  makes 
him  a  "  measurer "  ;  he  must  think  how  distant  the 
hunting  ground  is,  how  long  he  will  be  gone,  how 
much  he  should  carry  with  him,  etc.  If  he  has  a 
choice  of  ways  of  reaching  his  hunting  ground,  the 
numerical  valuation  becomes  more  marked ;  short  and 
long,  near  and  far  become  more  closely  defined.  The 
comparison  of  dift'erent  means  as  to  tbeir  serviceable- 


II  I' 


I  f 


r  I 


1^ 


40 


THE  PSYCHOLOGY  OF  NUMBER. 


ness  in  reaching  an  end  not  only  gives  ns  a  vague  idea 
of  their  quantity,  but  tends  to  make  it  precise,  nu- 
merical. 

The  importance  of  this  process  of  comparing  differ- 
ent means  in  order  to  select  the  best,  in  the  develop- 
ment of  number  judgments,  may  be  illustrated  as  fol- 
lows :  A  chick  just  out  of  the  shell  will  peck  accurately 
at  a  grain  of  corn  or  at  a  Hy.  We  might  say  that  it 
measured  the  distance.  But  this  does  not  mean  that  it 
has  any  idea  of  distance  or  that  there  is  any  conscious 
process  of  estimating  its  extent.  There  is,  in  reality, 
no  measuring,  no  comparison,  no  selection,  but  simply 
direct  response  to  the  stimulus ;  and  there  is,  therefore, 
no  sense  of  distance.  So  an  average  child  by  the  time 
he  is  six  months  old  will  reach  out  only  for  objects 
which  fall  within  the  length  of  his  arms,  while  pre- 
viously he  may  have  attempted  to  grasp  objects  irre- 
spective of  their  distance.  Yet  he  does  not  measure, 
or  necessarily  have  a  consciousness  of  distance,  unless 
there  happen  to  be,  say,  two  objects,  one  just  within 
liis  reach,  the  other  just  beyond,  and  he  selects  the 
nearest  object  as  a  result  of  comparing  the  distances. 
If  the  child  does  this,  he  performs  a  rudimentary  meas- 
urement and  has  a  crude  idea  of  distance.  So,  too, 
creeping,  walking,  etc.,  imply  what  may  be  termed 
measurement,  but  they  involve  no  j^rocess  of  measur- 
ing, and  hence  no  consciousness  of  space  values,  of 
length,  or  size,  or  form,  until  the  child  begins  to  pre- 
fer and  select  one  of  several  paths.  AVHien  he  does  this, 
he  refers  the  various  paths  to  his  own  ease  of  aetumj 
and  thus  gets  a  standard  for  comparison. 

Suinntary. — The  conscious  adjusting  of   means  to 


THE  ORIdlN  OF   NUMBER. 


41 


end,  particularly  such  an  adjusting  as  requires  compari- 
son of  different  means  to  pick  out  the  fittest,  is  the 
source  of  all  quantitative  ideas — ideas  such  as  more  and 
less,  nearer  and  farther,  heavier  and  lighter,  etc.  Quan- 
tity means  the  valuation  of  a  thing  with  reference  to 
some  end  ;  what  is  its  icortli^  its  effectiveness^  compared  , 
with  other  iwssible  means.  These  two  conceptions — 
{a)  the  origin  of  quantitative  ideas  in  the  process  of 
valuation  (measuring)  and  ijj)  the  dependence  of  valu- 
ation upon  the  adjusting  of  means  to  an  end  (i.  e., 
ultimately  upon  activity)  are  the  beginning  of  all  con- 
ceptions of  quantity  and  number,  and  the  sound  basis 
of  all  dealing  with  them. 

The  Idea  of  Balance  ok  Equation. — We  shall  now 
note  more  definitely  what  is  implied  in  the  foregoing 
account — the  cont-tant  aiming  at  a  balance  or  equiva- 
lence— {valeas,  worth  ;  mjuns,  equal).  The  process  of 
adjusting  means  to  end  is  not  simply  a  process  of 
roughly  estimating  the  value  of  certain  things  with  re- 
gard to  the  end  aimed  at ;  but,  as  already  said,  it  is 
economical  and  successful  in  the  deiijree  in  which  is 
employed  just  the  auiount  of  energy,  just  the  amount 
of  means  necessary  to  accomplish  the  aim.  The  means 
used  must  just  balance  the  end  sought.  Every  machine, 
for  example,  represents  an  adjustment  of  certain  means 
to  a  certain  end.  But  there  are  good  machines  and  bad 
machines.  What  constitutes  the  difference  between  a 
good  machine  and  a  poor  one  ?  The  difference  is  found 
precisely  in  the  fact  that  the  former  represents  in  itself 
and  in  the  arrangement  of  its  parts  not  only  an  adjust- 
ment in  general  to  sofne  end,  but  an  accurate  adjustment 
to  the  jn'ecise  end  to  be  reached  ;  it  is  the  embodiment 


iiil 


'I  'i 
1  )'■ 


II:. 


(i 


!•  I 


i 


42 


THE  PSYCHOLOGY  OF  NUMBER. 


|1 


i    ' 


in  wood  and  iron  of  a  series  of  meclianical  principles — it 
represents  an  equation.  Now  it  is  this  necessity  of  ex- 
act balance  or  eqnivalency  which  transforms  the  vague 
quantitative  ideas  of  smaller  and  greater,  heavy  and 
light,  and  so  on,  into  the  definite  quantitative  ideas  of 
just  so  distant,  just  so  long,  so  heavy,  so  elastic,  etc. 
This  demands  tlie  introduction  of  the  idea  of  inimher. 
Numljer  is  the  delinite  measurement,  the  definite  valu- 
ation of  a  quantity  falling  within  a  given  limit.  Ex- 
cept as  we  count  off  means  and  end  into  just  so  many 
definite  units,  there  can  not  be  an  economical  adjust- 
ment, and  there  can  not  be  a  precise  balance,    y 

Summary. — Number  arises  in  the  process  of  the 
exact  measurement  of  a  given  quantify  with  a  view  to 
instituting  a  balance,  the  need  of  this  balance,  or  accu- 
rate adjustment  of  means  to  end,  being  some  limitation. 

Illustration. — The  logical  steps  of  the  development 
of  number  may  be  illustrated  as  follows  :  First,  there  is 
a  recognition  that  one  is  distant  from  one's  destination 
— say,  the  camp.  Next,  that  one  can  by  travelling  fast 
reach  the  camp  by  sunset.  Third,  the  recognition  that 
the  present  time  (the  time  of  starting)  is  such  an  hour 
of  the  day,  e.  g.,  two  o'clock,  and  that  sunset  occurs  at 
such  a  time — say  seven  o'clock  ;  that  the  present  spot  is 
just  so  many  miles  distant  from  the  camp  ;  and  that, 
consequently,  one  will  have  to  travel  just  so  many  miles 
in  a  given  time — say  an  hour — to  reach  the  camp ;  this 
las!;  stage  being  the  equation  or  balance. 

The  Heasun  for  Abstraction  and  Generaliza- 
tion.— We  are  now  prepared  to  see  the  reason  for  the 
neglect  of  the  sense  qualities  (the  abstraction)  and  for 
the  reference  to  the  whole  (the  generalization)  included 


THE  ORIGIN  OF  NUMBER. 


43 


in  all  numbering.  When  we  are  regarding  a  tiling  not 
in  itself,  but  simply  as  a  meaiis  for  some  end,  we  take 
no  account  of  any  qualities  which  it  may  possess  except 
this  one  quality  of  being  related  to  the  end.  If  I  tnu 
to  find  out  merely  the  quantity  of  land  in  a  field,  tlie 
fact  that  a  part  of  the  field  is  heavy  clay  and  the  rest 
rich,  loamy  soil  is  not  taken  into  consideration  ;  these 
qualities  do  not  make  the  size  value  of  the  field,  and  are 
nothing  to  my  purpose.  I  restrict  attention  entirely  to 
the  rnatheiiiatical  measurements,  which  in  themselves 
are  necessarv  and  sufficient  for  the  end  to  be  reached — 
the  determination  of  the  absolute  area  of  the  field.  But 
if  I  am  to  compute  the  money  value  of  the  field,  and 
know  that  the  loamy  soil  has  one  value  and  the  clayey 
soil  another,  these  qualities,  having  a  relation  to  the  end 
in  view,  would  have  to  be  noted  as  controlling  the 
measurements ;  while  all  other  qualities — kind  of  clay, 
character  of  loam,  moisture,  and  dryness — would  be 
neglected  as  not  bearing  on  the  question  of  the  money 
value  of  the  field. 

Similarly,  if  we  want  to  know  the  whole  amount  of 
cloth  in  a  store,  we  neglect  all  special  qualities  of  cloth, 
and  abstract  the  one  quality  of  being  cloth  ;  silks,  wool- 
lens, linens,  cottons,  however  marked  their  differences, 
are  alike  in  possessing  this  one  quality  that  niakes  for 
our  present  purpose — they  are  all  cloth.  But  if  we  are 
required  to  find  the  total  money  value,  as  in  taking  an 
inventory,  and  the  different  kinds  of  cloth  have  diflierent 
prices,  tlien  we  should  abstract  the  special  quality  of  be- 
ing silk,  or  linen,  or  woollen,  or  cotton,  and  neglect  all 
other  qualities,  colours,  patterns,  etc.,  which  are  believed 
to  have  no  effect  upon  the  price.     In  other  words,  it  is 


ii    t 


11 

I) 


I    I 


: 


44 


THE  rSYCITOLOGY  OF  NUMBER. 


always  the  end  hi  v/efr  wliich  decides  wliat  qualities 
we  shall  pay  attention  to  and  what  netijlect.  We  ab- 
stract, or  select,  the  special  quality  that  /n'/j)s  with  i"ef- 
erence  to  this  end.  The  rest,  for  purposes  of  measure- 
ment, are  nothiuir  to  us. 

It  is  obvious  that  it  is  the  same  reference  to  the 
end  to  be  accomplished  that  constitutes  (/e7ieralhatwn. 
We  regard  the  various  objects  selected  as  having  a 
relation,  as  making  up  one  whole  or  class,  because,  no 
matter  what  their  differences  in  themselves,  they  all 
serve  the  same  end.  It  is  this  common  service  in 
helping  towards  one  and  the  same  end  which  binds 
them  together,  even  if  to  eye  or  ear  the  things  are 
entirely  different.  It  is  the  reference  to  the  end  to  l)e 
reached  that  controls  both  the  abstraction  and  the  gen- 
eralization. 

The  books  composing  a  library  may  be  of  many 
kinds — primers  and  dictionaries,  novels  and  poems, 
printed  in  all  languages,  with  pages  of  all  sizes,  and 
bindings  in  endless  variety,  yet  as  serving  the  one  pur- 
pose of  communicating  intelligence  through  written  or 
printed  symbols  they  all  fall  together,  and  can  be  count- 
ed as  making  up  one  group. 

The  Process  of  Measuring. 

We  have  now  (1)  noted  the  psychological  processes 
of  abstraction  and  generalization  involved  in  all  num- 
ber, and  have  (2)  traced  them  to  the  r.eed  of  an  eco- 
nomical adjustment  of  means  to  end  which  makes  neces- 
sary the  process  of  measuring  from  which  mimher  has 
its  genesis.  We  have  now  to  note  in  more  detail  the 
nature  of  this  latter  process. 


THE  OlllUIN   OF  NUMBER. 


45 


Stacjes  of  Measurement. — AVc  begin  with  the  vague 
estimate  of  bulk,  size,  weight,  etc.,  and  go  on  to  its  ac- 
curate determination  from  the  indefinite  how  much  to 
the  definite  so  much.  It  is  the  diiference  between  say- 
ing tliat  iron  is  heavy  and  that  so  much  iron  at  a  given 
temperature  and  a  given  latitude  weighs  just  so  much  ; 
or  between  saying  that  the  blackboard  is  of  modei-ate 
size  and  that  it  contains  so  many  square  feet.  The  de- 
velopment from  the  crude  guess  to  the  exact  statement 
depends  upon  the  selection  and  recognition  of  a  uiut^ 
the  repetition  of  which  in  space  or  time  makes  up  and 
thus  measures  the  whole.  The  savage  may  begin  by 
saying  that  his  camp  is  so  many  suns  away.  Here 
his  unit  is  the  distance  he  can  travel  between  sunrise 
and  sunset.  lie  measures  by  a  unit  of  action,  but  that 
unit  is  itself  unmeasured — just  how  much  it  is  he  can 
not  say — or  he  measures  by  marking  off  so  many  paces. 
The  pace  is  the  unit,  and  is,  relatively,  more  definite  or 
accurate  than  the  day's  journey,  but,  absolutely,  it  is 
unmeasured.  Only  when  the  unit  itself  is  accurately 
defined  do  we  pass  from  vague  quantity  to  precise  nu- 
merical value. 

Quantities  in  Different  Scales  of  Measurement. — 
But  the  process  of  measurement  may  be  carried  a  step 
further.  For  accurate  measurement  the  unit  itself  must 
be  measured  with  a  unit  of  the  same  kind  of  quantity ; 
but  the  unit  may  also  have  a  defined  relation  to  a  dif- 
ferent kind  of  quantity.  We  are  thus  enabled  to  com- 
pare quantities  lying  in  differe7it  scales  of  measurement. 
We  can  not,  for  example,  directly  compare  weights  and 
volumes,  but  we  may  compare  them  indirectly.  If  we 
discover,  for  instance,  that  four  cubic  inches  of  iron 


i !! 


46 


TlIK   PSYCIIOLOGV   OF  NUMBER. 


weigli  as  much  as  twenty-nine  cubic  inclies  of  water, 
four  cubic  inches  of  gold  as  much  as  seventy-seven 
cubic  inches  of  water,  and  two  cubic  inches  of  lionev  as 
much  as  three  cubic  inches  of  water,  we  have  then  the 
means  of  comparing  cubic  inclies  of  iron  with  penny- 
weights of  gold  or  with  pounds  of  honey.  Thus  the 
different  scales  of  volume  and  weight  are  brought  to- 
gether by  comparing  both  with  a  common  standard. 
Take  a  case  of  comparing  money  values.  AVe  can 
directly  compare  the  cost  of  three  yards  of  calico  with 
that  of  nine  yards  of  the  same  quality  ;  but  we  can  not 
directly  compare — as  to  cost — lengths  of  calico  of  dif- 
ferent qualities,  or  a  length  of  calico  with  one  of  silk. 
But  if  we  know  that  one  yard  of  calico  is  worth  (is 
measured  by)  eight  cents,  and  one  yard  of  silk  worth 
one  dollar  and  sixty  cents,  we  can  accurately  measure 
the  worth  of  any  quantity  of  calico  in  terms  of  any 
quantity  of  silk. 

If  it  were  not  for  this  discovery  of  a  unit  differing 
in  kind  from  the  quantity  to  be  measured,  and  yet  capa- 
ble of  comparison  with  it,  our  exact  measurements 
would  always  be  confined  within  one  and  the  same 
scale — time,  weight,  volume,  etc. ;  we  should  simply 
have  to  guess  how  much  of  one  scale  would  equal  a 
given  quantity  of  another. 

Moreover,  the  measurement  within  a  given  scalr  '<♦ 
imperfect  if  we  have  no  means  of  defining  some  unit  . 
the  scale  in  terms  of  a  different  quantity.  AYe  may 
know  how  much  a  pound  is  in  terms  of  an  ounce,  an 
ounce  in  terms  of  drachms,  but  we  can  never  get  out  of 
this  circle.  We  can  never  know  how  much  the  ounce, 
the  pound,  etc.,  really  is ;  our  measurement  can  not 


I'  ' 


THE  ORIGIN  OP  NUMBER. 


47 


i 


reach  the  highest  stage  of  development — what  may  bo 
called  the  scientific  stage. 

There  is  perfect  measurement  only  when  this  stage 
is  reached.  The  pound  of  the  weight  scale,  for  exam- 
ple, is  not  perfectly  defined  in  terms  of  weight  (ounces, 
etc.)  alone ;  the  pound  is  more  accurately  defined  when 
we  discover  that  it  is,  say,  the  amount  of  copper  which 
under  certain  conditions  will  displace  such  and  such  an 
amount  of  water.  Only  in  some  such  way  as  this  is  our 
unit  ultimately  defined,  and  only  when  the  unit  of 
measure  is  itself  perfectly  measured  can  there  be  per- 
fectly exact  or  scientific  measurement.  This  measure- 
ment of  a  quantity  in  terms  of  quantity  unlike  in  kind, 
but  alike  in  some  one  respect,'^  is  the  completion  of 
number  as  the  tool  of  measurement.  Beyond  this  stage, 
number  can  not  go,  but  until  it  has  developed  to  this 
])oint  it  is  an  imperfect  instrument  of  measurement. 
There  are  therefore  three  stages  of  measurement : 

1.  Measuring  with  an  undefined  unit,  as  in  measur- 
ing length  by  the  unit  "  pace,"  apples  by  the  unit  ap- 
ple, etc. 

2.  Measuring  with  a  unit  itself  defined  by  compari- 
son with  a  unit  of  same  kind  of  quantity — the  yard,  the 
pound,  the  dollar,  etc. 

3.  Measuring  with  a  unit  having  a  definite  relation 
to  a  quantity  of  a  different  kind. 

Cou7itin(j  and  Measuring. — It  has  been  said  that 
number  originates  from  measurement ;  that  it  is  a  state- 
mf'nt  of  the  numerical  value  of  something.  But  we  are 
8'    iistomed  to  distinguish  counting  (i.  e.,  numeration, 

*  This  common  basis  of  comparison  is  always,  ultimately,  move- 
K    nt  in  space. 


^1  b 


;l    I 


'■|: 


I 

III' 


I 


OQ 


I 


i 


jii 


111 


?! 


48 


THE  PSYCHOLOGY  OF  NUMBER. 


numbering)  from  measuring.  It  is  usually  said  that  we 
count  objects,  particular  things  or  qualities,  to  see  how 
many  of  them  there  are,  while  we  measure  a  particular 
object  or  quality  to  see  how  much  of  it  there  is.  We 
count  chairs,  beds,  splints,  feet,  eyes,  children,  stamens, 
etc.,  simply  to  get  their  sum  total,  the  lioio  many  '^  we 
measure  distance,  weight,  bulk^  ])rice,  cost,  etc.,  to  see 
Iwio  much  there  is.  Some  writers  say  that  these  "  cwo 
kinds  "  of  quantity,  which  they  call  quantity  of  magni- 
tude (how  much)  and  quantity  of  multitude  (how  many), 
^  are  entirely  distinct.(^  Nevertheless,  all  counting  is 
1  measuring,  and  all  mekglti-iTrg  is  counting.  ,  When  we 
count  up  the  number  of  particular  books  in  a  library, 
we  measure  the  library — iind  out  how  much  it  amounts 
to  as  a  library ;  when  we  count  the  days  of  tlio  year,  we 
measure  the  time  value  of  the  year ;  when  we  count  the 
children  in  a  class,  we  measure  the  class  as  a  whole — it 
is  a  large  or  a  small  class,  etc.  When  we  count  stamens 
or  pistils,  we  measure  the  flower.  In  short,  when  we 
COUNT  we  measure. 

On  the  other  hand,  in  measuring  a  continuous  quan- 
tity— "quantity  of  magnitude" — counting  is  equally 
necessary.  We  may  apply  a  unit  of  measure  to  such  a 
quantity  and  mark  off  tlic  parts  with  perfect  accuracy, 
but  there  is  no  measurement  till  we  have  couyited  the 
])firts.  Thus,  the  only  way  to  measure  weight  is  by 
counting  so  many  iin'ds  of  density ;  distance,  by  count- 
ing so  many  particular  units  of  length  ;  cost  or  price,  by 
counting  so  many  units  of  value — dollars  or  wha:  not. 
In  other  words,  when  we  measure  we  count.  The  dif- 
ference is  that  in  what  is  ordinarily  termed  counting,  as 
distinct  from  measuring,  we  work  with  an  undefined 


I 


!    W 


THE  ORIGIN  OF  NUMBER. 


49 


unit ;  ■  t  is  vague  measurement,  because  our  unit  is  un- 
measured. When  we  say  ten  apples,  live  books,  six 
horses,  etc.,  we  measure  some  whole,  some  how  7)iuch, 
by  counting  its  parts,  the  how  many  /  but  we  do  not 
know  just  how  much  one  of  these  parts  or  units  is.  If 
we  knew  the  exact  size,  or  weight,  or  price  of  the  books 
or  apples,  we  should  have  a  more  accurate  measurement 
and  a  more  accurate  valuation.*  On  the  o  \er  hand, 
what  we  ordinarily  call  measuring,  as  distinct  from 
counting,  is  simply  counting  with  a  unit  which  is  itself 
measured  by  so  many  deiinite  parts.  If  I  count  oif 
four  books,  "  book,"  the  unit  which  serves  as  unit  of 
measurement,  is  itself  only  a  qualitative,  not  a  quanti- 
tative unity,  and  the  quantity  four  books  is  not  a  defi- 
nitely measured  quantity.  If  I  say  each  book  weighs  six 
ounces  or  is  worth  sixty  cents,  the  unit  of  measurement 
is  itself  both  qualitative  and  quantitative ;  and  the  price 
or  the  weight  of  the  four  books  is  a  definitely  measured 
quantity. 

We  shall  see  hereafter  that,  strictly  speaking,  merely 
qualitative  wholes  used  as  units  give  only  addition  and 
subtraction  ;  that  the  whole  which  is  itself  quantitative, 
as  well  as  qualitative,  gives  multiplication  and  division. 
If,  however,  the  wholes  are  taken  or  assumed  as  equal 
in  value,  then,  of  course,  the  operations  of  multiplica- 
tioTi  and  division  may  be  performed  with  them.  But 
this  is  only  because  the  assumption  of  equal  (measured) 

*  It  is  a  great  pity  that  our  authorities  use  these  unmeasured 
units  so  much,  particularly  in  fractions.  Half  an  apple,  half  a  pie, 
is  a  practical,  not  a  mathematical  expression  at  all.  To  make  it 
mathematical  we  should  have  to  know  just  how  great  the  whole  is — 
how  many  ounces  or  cubic  inches. 


;  ii 


^:i 


iPi 


i 


■..-*w:-'-->.  .■»;'«ij;iiJ?Ht,iivcr»fi-ri. ■-■*■■•' 


60 


THE  PSYCHOLOGY  OF  NUMBER. 


f 


* 


value  in  the  units  is  made.  If  we  are  to  divide  fifteen 
apples  "equally"  among  five  boys,  giving  each  boy 
three  apples,  this  "equal"  distribution  assumes  the 
eqicality  of  the  units  (apples)  of  measure. 

Much  and  Many. — The  whole  falling  within  a  cer- 
tain limit  supplies  the  muchness  j  for  example,  the 
amount  of  money  in  a  purse,  the  amount  of  land  in  a 
field,  the  amount  of  pressure  it  takes  to  move  an  obsta- 
cle, etc.  This  "  much,"  or  amount,  is  vague  and  unde- 
fined till  measured ;  it  is  measured  by  counting  it  off 
into  so  many  units.  We  "lay  off"  distance  into  so 
many  yards,  and  then  we  know  it  to  be  so  much.  We 
reckon  up  the  pieces  of  money  in  the  purse  and  know 
how  much  their  value  is.  A  man  has  a  pile  of  lumber; 
liow  "  much  "  has  he  ?  If  the  boards  are  of  uniform 
size,  he  finds  the  number  (how  many)  of  feet  in  one 
board,  ?iA  counts  the  number  (how  many)  of  boards, 
and  finds  the  -whole  soma7\y  feet — that  is,  the  indefinite 
"  how  much  "  has  become,  through  counting,  the  definite 
"  so  much."  Then,  again,  if  he  wishes  to  find  the  money 
value  of  the  lumber,  how  much  it  is  worth,  he  must 
count  off  the  total  number  of  feet  at  so  much  (so  many 
dollars)  per  thousand,  and  the  resulting  so  many  dollars 
represents  the  worth  of  the  lumber.  The  many,  the 
counting  up  of  the  particular  units,  measures  the  worth 
of  the  whole.  The  counting  has  no  other  meaning, 
and  the  measurement  of  value  can  occur  in  no  other 
way. 

It  is  clear  that  these  two  sides  of  all  number  are  rel- 
ative to  each  other,  just  as  means  and  ends  are  relative. 
The  so  many  measures  the  so  much,  just  as  the  means 
balance  the  end.     The  end  is  the  whole,  all  that  comes 


m 


« 


THE  ORIGIN   OF  NUMBER. 


61 


witliin  a  certain  limit ;  the  means  are  tlie  partial  activi- 
ties, the  units  by  which  we  reaUze  this  whole. 

To  build  a  house  of  a  certain  kind  and  value,  we 
must  have  just  so  many  bricks,  so  many  cubic  feet  of 
stone,  so  much  lumber,  so  many  days'  work,  etc.  The 
house  is  the  end,  the  goal  to  be  reached ;  these  things 
are  the  means.  The  house  has  been  erected  at  a  certain 
cost ;  the  counting  off  and  valuing  of  the  units  which 
enter  into  these  different  factors,  is  the  only  way  to  dis- 
cover that  cost. 


i. 


CHAPTER  lY. 

THE   ORIGIN   OF   NUMBEK  I    SUAiMAKY   AND   AITLICATIONS. 

SuM^iAKY  :  Complete  Activity  and  Subordinate  Acts. 
— Through  the  forciroino:  illustrations — wliicli  are  illiis- 
tratioiis  of  one  and  the  same  principle  regarded  from 
different  points  of  view — we  are  now  prepared  for  the 
statement  which  sums  up  this  preliminary  examination 
of  quantity.  21(at  whicJt  fixes  the  magnitude  or  quan- 
tity u'hicJt,  in  any  given  case^  needs  to  he  measured  is 
sofne  activity  or  movement^  internally  co?itinuous,  hut 
externally  limited.  Tliat  which  measures  this  whole 
is  some  ntinor  or  partial  activity  into  which  the  orig- 
inal continuous  activity  may  he  hrolicn  up  (analysis), 
and  which  repeated  a  cert<(in  numher  of  times  gives 
the  s((iiie  result  {synthesis)  as  the  original  continuous 
actirify. 

This  formula,  embodying  the  idea  that  number  is  to 
be  traced  to  measurement,  and  measurement  back  to  ad- 
justment of  activity,  is  the  key  t  the  entire  treatment 
of  number  as  presented  in  tliese  pages,  and  the  reader 
should  be  sure  lie  understands  its  meaniui:^  before  ffoinji: 
furtlier.  In  order  to  test  his  comprehension  of  it  he 
may  ask  himself  such  questions  as  these  :  The  year  is 
some  unified  activity — what  activity  does  it  represent  ? 
At  first  sight  simply  the  apjiarent  return  of  the  sun  to 


,  1 


THE  ORIGIN  OF  NUMBER. 


63 


the  same  point  in  the  heavens — a:i  external  change  ;  yet 
the  only  reason  for  attaching  so  much  importance  to 
this  rather  than  to  any  other  cyclical  change,  as  to 
make  it  the  unit  of  time  measurement,  is  that  the 
movement  of  the  sun  controls  the  cycle  of  human  ac- 
tivities— from  seedtime  to  seedtime,  from  harvest  to 
harvest.  This  is  illustrated  historically  in  the  fact  that 
until  men  reached  the  agricultural  stage,  or  else  a  con- 
dition of  nomadic  life  in  which  their  movements  were 
controlled  by  the  movement  of  the  sun,  they  did  not 
take  the  sun's  movement  as  a  measure  of  time.  So, 
again,  the  day  represents  not  simply  an  external  change, 
a  recurrent  movement  in  IS^ature,  but  a  rhythmic  cycle 
of  human  action.  Again,  what  activity  is  represented 
by  the  pound,  by  the  bushel,  by  the  foot  ?  '-^  What  is 
the  connection  between  the  decimal  system  and  the  ten 
fingers  of  the  hands  ?  What  activity  does  the  dollar 
stand  for  ?  If  the  dollar  did  not  represent  certain  pos- 
sible activities  which  it  places  at  our  control,  would  it 
be  a  measure  of  value  ?  AVhy  may  a  child  value  a  bright 
penny  higher  than  a  dull  dollar  ?     And  so  on. 

Illustrations:  Stages  of  Measurement. —  Suppose 
we  w4sh  to  find  the  quantity  of  land  in  a  cei'tain 
field.  The  eye  runs  down  the  length  and  along  the 
breadth  of  the  field  ;  there  is  the  sense  of  a  certain 
amount  of  movement.  This  activity,  limited  by  the 
boundaries  of  the  field,  constitutes  the  original  vague 
muchness — the  quantity  to  be  measured — and  therefore 
determines  all  succeeding  processes.  Then  analysis 
comes  in,  the  breaking  up  of  this  original  continuous 

*  The  historical  origin  of  these  measures  will  throw  light  upon 
the  psychological  point. 


V  i 


rrr:'!'i*  <'  Jt-ff"! 


54 


THE  PSYCnOLOGY  OF  NUMBER.- 


activity  into  a  series  of  minor,  discrete  acts.  The  eye 
runs  down  the  side  of  the  field  and  fixes  upon  a  point 
wliich  appears  to  mark  half  the  length ;  this  process  is 
repeated  with  each  half  and  with  each  quarter,  and  thus 
the  length  is  divided  roughly  into  eight  parts,  each 
roughly  estimated  at  twenty  paces.  The  breadth  of  the 
field  is  treated  in  the  same  way.  The  eye  moves  along 
till  it  has  measured,  as  nearly  as  w^e  can  judge,  just  as 
much  space  as  equals  one  of  the  smallest  divisions  on 
the  other  side. 

The  process  is  repeated,  and  we  estimate  that  the 
breadth  contains  six  of  these  divisions.  Through  these 
interrupted  or  discrete  movements  of  the  eye  we  are 
able  to  form  a  crude  idea  of  tbe  length  and  breadth  of 
the  field,  and  thus  make  a  rough  estimate  of  its  area. 
The  separate  eye  movements  constitute  the  analysis 
w^hicli  gives  the  unit  of  measurement,  and  the  counting 
of  these  separate  movements  (units)  is  the  synthesis 
giving  the  total  numerical  value. 

But  the  breaking  up  of  the  original  continuous 
movement  into  minor  units  of  activity  is  obviously 
crude  and  defective,  and  hence  the  resulting  syntliesis 
is  imperfect  and  inadequate.  The  only  thing  we  are 
certain  of  is  the  number  of  times  the  minor  act  has  been 
performed ;  it  is  pure  assumption  tliat  the  minor  act 
measures  an  equal  length  every  time,  and  a  mere  guess 
that  each  of  the  lengths  is  twenty  yards.  In  order, 
therefore,  to  make  a  closer  estimate  of  the  content  of 
the  field,  we  may  mark  off  the  length  and  breadth  by 
pacing,  and  find  that  it  is  a  hundred  and  seventy  paces 
in  length  and  a  hundred  and  thirty  ])aces  in  breadth. 
This  is  probably  a  more  correct  estimate,  because  {a)  wo 


THE  ORIGIN  OF  NUMBER. 


55 


can  be  much  more  certain  that  the  various  paces  are 
practically  equivalent  to  one  another  than  that  the  eye 
movements  are  equal,  and  (h)  since  the  pace  is  a  more 
detinite  and  controlled  movement,  we  have  a  much 
clearer  idea  of  how  much  the  pace  or  unit  of  measure- 
ment really  is. 

J3ut  it  is  still  an  assumption  that  the  various  paces 
are  equal  to  one  another.  In  other  words,  this  unit  of 
measure  is  not  itself  a  constant  and  measured  thing,  and 
the  required  measurement  is  therefore  still  imperfect. 
Hence  the  substitution  for  the  pace  of  some  measuring 
unit,  say  the  chain,  which  is  itself  defined ;  the  chain  is 
applied,  laid  down  and  taken  up,  a  certain  number  of 
times  to  both  the  length  and  the  breadth  of  the  field. 
Now  the  minor  act  is  uniform  ;  it  is  controlled  by  the 
measuring  instrument,  and  hence  marks  off  exactly  the 
same  sjMce  every  thne!^  The  partial  activity  being  de- 
fined, the  resulting  numerical  value — say,  eight  chains  by 
six  chains — is  equally  definite.  Besides,  the  chain  itself 
may  be  measured  off  into  a  certain  number  of  equal 
portions ;  we  may  apply  a  minor  unit  of  measure — e.  g., 
the  link — until  we  have  determined  how  many  links 
make  up  the  chain.  By  means  of  this  analysis  into  still 
smaller  acts,  the  meaning  of  the  unit  is  brought  more 
definitely  home  to  consciousness.f 

*  Note  how  the  two  factors  of  space  and  time  appear  in  all  meas- 
nrenient,  .s;>rtce  representing  concrete  value,  time,  the  abstract  num- 
ber, and  both,  the  measured  maji^nitude. 

f  If  it  be  noted  that  all  we  have  done  here  is  to  make  the  original 
activity  of  running  the  eye  along  length  and  breadth  first  continu- 
ously and  then  in  an  interrupted  series  of  minor  movements,  more 
controlled  and  hence  more  precise,  the  meaning  of  the  proposition 
(page  52)  regarding  the  origin  of  measurement  in  the  adjustment  of 


n 


i 


.,1 


56 


THE  PSYCHOLOGY  OP  NUiMBER. 


u 


But  tliis  mathematical  measurement,  this  analysis- 
synthesis,  is  still  insufficient  for  complete  adjustment  of 
activity.  What,  after  all,  is  the  value  of  this  measured 
quality  ?  What  is  it  good  for  ?  Until  this  question  is 
answered  there  can  not  be  perfect  adjustment  of  activi- 
ties. To  answer  this  brings  us  to  the  third  and  final 
stage  of  number  measurement.  This  field  will  produce, 
say,  only  so  many  bushels  of  corn  at  a  given  price  per 
bushel ;  it  is,  therefore,  not  worth  so  much  as  a  smaller 
field  which  will  produce  as  much  wheat  at  a  larger  price 
per  bushel.  Or,  in  addition  to  the  mere  size  of  the 
field,  it  may  be  necessary  to  take  into  account  not  only 
the  value  of  the  crop  it  will  raise,  but  also  the  cost  of 
tilling  it.  Here  there  must  be  a  much  more  complete 
adjustment  of  activities.  The  analysis  concerns  not 
only  so  many  square  rods ;  it  includes  also  the  money 
value  of  the  crop  and  the  cost  of  its  production.  The 
synthesis  compares  the  result  of  this  complex  measure- 
ment with  the  results  of  other  possible  distributions  of 
energy.  Analytically  the  conditions  are  completely  de- 
fined ;  synthetically  there  can  be  a  complete  and  eco- 
nomical adjustment  of  the  conditions  to  secure  the  best 
possil)le  results. 

The  measured  quantity  representing  the  unified  (or 
continuous)  activity  is  the  whole  or  unity ;  the  measur- 
ing parts,  representing  the  minor  or  partial  activities, 
are  the  components  or  units,  which  make  up  the  unified 
whole.  In  all  measurement  each  of  these  measuring 
parts  in  itself  is  a  w^hole  act — as  a  pace,  a  day's  journey, 
etc.     But  in  its  function  of  measuring  unit  it  is  at  once 


minor  acts  to  constitute  a  comprehensive  activity  will  be  apparent 
once  more. 


THE  ORIGIN  OF  NUMBER. 


67 


reduced  to  a  mere  means  of  constructing  tlie  more  com- 
prehensive act.  The  end  or  whole  is  one^  and  yet  made 
up  of  nnany  parts. 

Summary. — All  numerical  concepts  and  processes 
arise  in  the  process  of  fitting  together  a  number  of 
minor  acts  in  such  a  way  as  to  constitute  a  complete 
and  more  comprehensive  act. 

1.  This  fitting  together,  or  adjusting,  or  balancing, 
will  be  accurate  and  economical  just  in  the  degree  in 
which  the  minor  acts  are  the  same  in  Icincl  as  the  major. 
If,  for  example,  one  is  going  to  build  a  stone  wall,  the 
use  of  the  means — the  minor  activities — will  not  be  ac- 
curate until  one  can  find  a  common  measure  for  both 
the  means,  the  use  of  the  particular  stones,  and  the  end, 
the  wall.  Size,  or  amount  of  space  occupied,  is  this 
common  element.  Hence,  to  define  the  process  in  terms 
of  just  so  many  cubic  feet  recpiired  is  economical ;  to 
describe  it  in  terms  of  so  many  stones  would  be  impos- 
sible unless  one  had  first  found  the  volumes  of  the 
stones.  Hence,  once  more,  the  abstraction  and  the 
generalization  involved  in  all  numerical  processes — the 
special  qualities  of  the  stone  are  neglected,  and  the 
only  thing  considered  is  the  number  of  cu})ic  feet  in 
the  stone — abstraction.  But  through  this  factor  of 
so  much  size  the  stone  is  referred  at  once  to  its  place 
in  the  whole  wall  and  to  the  other  stones — general- 
ization. 

2.  An  end,  or  whole  of  a  certain  quality^  furnishes 
the  limit  within  which  the  magnitude  lies.  Quantity 
is  limited  quality,  and  there  is  no  quantity  save  where 
there  is  a  certain  qualitative  'whole  or  l/wiitation.      --' 

3.  Niimher  arises  through   the  use  of  means,  or 


I" 

III  *i 


ill 


'Ml 


a  1  \ 


68 


THE  PSYCHOLOGY  OF  NIjMBER. 


H 


(  ! 


M 


minor  units  of  activity,  to  construct  an  activity  equal 
in  value  to  the  given  magnitude..  This  process  of  con- 
structing an  equivalent  value  is  7nimheri}Hj — evaluation. 
Hence,  there  are  no  mtmerical  dlstmctlons  (psycho- 
logically) except  in  the  process  of  measuring  some 
qualitative  whole.* 

4.  This  measuring  or  valuation  (defining  the  original 
vague  qualitative  whole)  will  transform  the  vague  quan- 
tity into  precise  numerical  value  j  it  will  accomplish 
this  successfully  in  just  the  degree  in  which  the  minor 
activity  or  unity  of  construction  is  itself  measured,  or 
is  also  a  numerical  value.  Uidess  it  is  itself  a  numerical 
quantity,  a  unity  measured  by  being  counted  out  into  so 
many  parts,  the  minor  and  the  comprehensive  activity 
can  not  be  made  precisely  of  the  same  kind.  (Prin- 
ciple 1.) 

5.  Hence  the  purely  correlative  character  of  much 
and  many,  of  measured  whole  and  measuring  part,  of 
value  and  number,  of  unity  and  units,  of  end  and 
means. 

>1        Educational  Applications. 

We  have  now  to  apply  the  principle  concerning  the 
psychological  origin  of  quantity  and  number  to  educa-  ^ 
tion.  We  have  seen  {a)  the  need  in  life,  the  demand  in 
actual  experience  of  the  race  and  the  individual,  w^hich 
brings  the  numerical  operations ;  the  process  of  meas- 
uring, into  existence.  We  have  seen  {b)  what  forms 
number  is  required  to  assume  in  order  to  meet  the  need, 
fulfil  the  demand.     We  have  now  to  inquire  how  far 


*  The  pedagogical  consequences  of  neglecting  this  principle  will 
be  seen  in  discussing  the  Grube  method,  or  use  of  iha  fixed  unit. 


( 

•     1 


THE  ORIGIN  OP  NUMBER. 


50 


these  ideas  and  principles  have  a  practical  application 
in  educational  processes. 

The  school  and  its  operations  must  be  either  a  nat- 
ural or  an  artificial  thing.  Every  one  will  admit  that 
if  it  is  artificial,  if  it  abandons  or  distorts  the  normal 
processes  of  gaining  and  using  experience,  it  is  false  to 
its  aim  and  inefiicient  in  its  method.  The  development 
of  number  in  the  schools  should  therefore  follow  the 
principle  of  its  normal  psychological  development  in 
life.  If  this  normal  origin  and  growth  have  been  cor- 
rectly described,  we  have  a  means  for  determining  the 
true  place  of  number  as  a  means  of  education.  It  will 
require  further  development  of  the  idea  of  number  to 
show  the  educational  principles  corresponding  to  the 
growth  of  numerical  concepts  and  operations  in  them- 
selves, but  we  already  have  the  principle  for  deciding 
how  number  is  to  be  treated  as  regards  other  phases  of 
experience. 

The  Two  Methods  :  Things  ;  Symbols. — The  prin- 
ciple corresponding  with  the  psychological  law — the 
translation  of  the  psychological  theory  into  educational 
practice — may  be  most  clearly  brought  out  by  contrast- 
ing it  with  two  methods  of  teaching,  opposed  to  each 
other,  and  yet  both  at  variance  with  normal  psycho- 
logical growth.  These  two  methods  consist,  the  one 
in  teaching  number  merely  as  a  set  of  syiiibols ;  the 
other  in  treating  it  as  a  direct  jproperty  of  objects. 
The  former  method,  that  of  symbols,  is  illustrated  in 
the  old-fashioned  ways — not  yet  quite  obsolete  —  of 
teaching  addition,  subtraction,  etc.,  as  something  to  be 
done  with  "  figures,"  and  giving  elaborate  rules  which 
might  guide  the  doer  to  certain  results  called  "  answers." 


I! 


!!' 


I" 


XT 


if. 


GO 


THE  PSYCHOLOGY   OP  NUMBER. 


il' 


I 


It  is  little  more  than  a  blind  manipulation  of  num- 
ber symbols.  The  child  siniply  takes,  for  example,  the 
figures  3  and  12,  and  performs  certain  "operations" 
with  them,  which  are  dignified  by  the  names  addition, 
subtraction,  multiplication,  etc. ;  he  know's  very  little 
of  what  the  figures  signify,  and  less  of  the  meaning  of 
the  operations.  The  second  method,  the  simple  percep- 
tion or  observation  method,  depends  almost  wholly  upon 
physical  operations  wdth  things.  Objects  of  various 
kinds — beans,  shoe-pegs,  splints,  chairs,  blocks — are  sepa- 
rated and  combined  in  various  ways,  and  true  ideas  of 
number  and  of  numerical  operations  are  supposed  neces- 
sarily to  arise. 

Both  of  these  methods  are  vitiated  by  the  same  fun- 
damental psychological  error ;  they  do  not  take  account 
of  the  fact  that  number  arises  in  and  through  the  ac- 
tivity of  mind  in  dealing  with  ohjccts.  The  first  meth- 
od leaves  out  the  objects  entirely,  or  at  least  makes  no 
reflective  and  systematic  use  of  them ;  it  lays  the  em- 
phasis on  symbols,  never  showing  clearly  wdiat  they 
symbolize,  but  leaving  it  to  the  chances  of  future  expe- 
rience to  put  some  meaning  into  empty  abstractions. 
The  second  method  brings  in  the  objects,  but  so  far  as 
it  emphasizes  the  objects  to  the  neglect  of  the  mental 
activity  which  uses  them,  it  also  makes  number  mean- 
ingless ;  it  subordinates  thought  (i.  e.,  mathcmfi.tical  ab- 
straction) to  things.  Practically  it  may  be  considered 
an  improvement  on  the  first  method,  because  it  is  not 
possible  to  suppress  entirely  the  activity  which  uses  the 
things  for  the  realization  of  some  end ;  but  whenever 
this  activity  is  made  incidental  and  not  important,  the 
method  comes  far  short  of  the  intelligence  and  skill 


THE  ORIGIN  OP  NUMBER. 


61 


that  should  be  had  from  instruction  hascd  on  psycho- 
logical principles. 

While  the  inetliod  of  syrnholft  is  still  far  too  widely 
used  in  practice,  no  educationist  defends  it ;  all  con- 
dejnii  it.  It  is  not,  then,  necessary  to  dwell  upon  it 
lunger  than  to  point  out  in  the  light  of  the  previous 
discussion  wiry  it  should  be  condemned.  It  treats  num- 
ber as  an  independent  entity — as  something  apart  from 
the  mental  activity  which  produces  it ;  the  natural  gen- 
esis and  use  of  nund)er  are  ignored,  and,  as  a  result, 
the  method  is  mechanical  and  artificial.  It  subordinates 
sense  to  symbol. 

The  method  of  things — of  observing  objects  and 
taking  vague  percepts  for  definite  numei'ical  concepts — 
treats  number  as  if  it  were  an  inherent  pro])erty  of 
things  in  themselves,  simply  waiting  for  the  mind  to 
grasp  it,  to  "abstract"  it  from  the  things.  I>ut  we 
have  seen  that  number  is  in  reality  a  mode  of  ?ne(isifr- 
ing  vahte^  and  that  it  does  not  belong  to  things  in  them- 
selves, but  arises  in  the  economical  adaptation  of  things 
to  some  use  or  purpose.  Numljer  is  not  (psychologic- 
ally) got  from  things,  it  is  put  i7ito  them. 

It  is  then  almost  equally  absurd  to  attempt  to  teach 
numerical  ideas  and  ])roeess  vnthovt  things,  and  to  teach 
them  simply  hy  things.  Numerical  ideas  can  be  nor- 
mally acquired,  and  numerical  operations  fully  mastered 
only  by  arrangements  of  things — that  is,  by  certain  acts 
of  mental  construction,  which  are  aided,  of  course,  by 
acts  of  physical  construction  ;  it  is  not  the  mere  percep- 
tion of  the  things  which  gives  us  the  idea,  but  the  em-  '• 
ploying  of  the  things  i7i  a  coiistructim  way. 

The  method  of  symbols  supposes  that  number  arises 


1  A 


i 


62 


THE  rSYCTTOLOGY  OF  NUMBRR. 


* 


i: 


wholly  as  a  matter  of  abstract  reasoning ;  tlie  metliod 
of  objects  snpposes  that  it  arises  from  mere  observation 
by  the  senses — that  it  is  a  property  of  things,  an  ex- 
ternal energy  just  waiting  for  a  chance  to  seize  upon 
consciousness.  In  reality,  it  arises  from  constructive 
(psychical)  activit^y,  from  the  actual  use  of  certain  things 
in  reaching  a  certain  end.  This  method  of  constructive 
use  unites  in  itself  the  principles  of  both  abstract  rea- 
soning and  of  definite  sense  observation. 

If,  to  help  the  mental  proc^ess,  small  cubical  blocks 
are  used  to  build  a  large  cube  with,  there  is  necessarily 
continual  and  close  observation  of  the  various  things  in 
their  quantita^'Vv.  aspects;  if  splints  are  used  to  inclose 
a  surface  with,  the  particular  splints  must  be  noted. 
Indeed,  ihis  observation  is  likely  to  be  closer  and  more 
accurate  than  that  in  wdiich  the  mere  observation  is  an 
end  in  itself.  In  the  latter  case  there  is  no  interest,  no 
purpose,  and  attention  is  laboured  and  wandering;  there 
is  no  aim  to  guide  and  direct  the  observation.  The 
observation  wdiich  goes  with  constructive  activity  is  a 
part  of  the  activity  ;  it  has  all  the  intensity,  the  depth 
of  excitation  of  the  activity ;  it  shares  in  the  interest  of 
and  is  directed  by  the  activity.  In  the  case  where  the 
observation  is  made  the  whole  thing,  distinctions  have  to 
be  separately  noted  and  separately  memorized.  There 
is  nothing  intrinsic  by  which  to  carry  the  facts  noted ; 
that  the  two  blocks  here  and  the  two  there  make  four 
is  an  iWten^al  fact  to  be  carried  by  itself  in  memory. 
]hit  when  the  two  sets  are  so  used  as  to  construct  a 
whole  of  a  certain  value,  the  fact  is  internal  •  it  is  part 
of  the  mind's  w\ay  of  acting,  of  seeing  a  definite  whole 
through  seeing  its  definite  parts.     Repetition  in  one 


THE  OUIGIN   OF  NUMBER. 


G3 


case  means  sini})ly  learning  by  rote  ;  in  the  other  ease, 
it  means  repetition  of  activity  and  formation  of  an  in- 
telh'gent  liabit. 

The  rational  factor  is  found  in  the  fact  that  the  con- 
structive activity  proceeds  upon  a  principle ;  the  con- 
struction follows  a  certain  regular  or  orderly  method. 
The  method  of  action,  the  wav  of  combinini:^  the  means 
to  reacli  the  end,  tlie  ])arts  to  make  the  whole,  is  rela- 
tion /  acting  according  to  this  relation  is  rational,  ana 
prepares  for  the  definite  recognition  of  reason,  for  con- 
sciously grasping  the  nature  of  the  operations,  Ka- 
tional  action  will  pass  over  of  itself  when  the  time  is 
ripe  into  abstract  reasoning.  The  habit  of  abstracting 
and  generalizing  of  analysis  and  synthesis  grows  into 
definite  control  of  thinking. 

Thk  Factors  in  Kational  Method. — In  more  de^ 
tail,  dealing  with  number  by  itself,  as  represented  by 
symbols,  introduces  the  child  at  an  early  stage  to  ab- 
stractions without  showing  how  they  arise,  or  what  they 
stand  for ;  and  makes  clear  no  reason,  no  /'^cessity,  for 
tlie  various  operations  performed,  which  are  all  reduci- 
ble to  {(()  synthesis — addition,  multi])lication,  involu- 
tion ;  and  (/>)  analysis — subtraction,  division,  evolution. 
The  object  or  observation  method  shows  the  relation 
of  number  to  things,  but  does  not  make  evident  why  it 
has  this  rchition  ;  does  not  bring  out  its  value  or  meas- 
uiing  use,  and  leaves  the  o})erations  performed  pui'ely 
external  manipulations  of  number,  or  rather  with  things 
which  may  be  numbered,  not  internal  developments  of 
its  measuring  power.  The  method  which  develops  nu- 
merical ideas  in  connection  with  ^he  construction  of 
eonio  definite  thing,  I/rings  out  clearly  {a)  the  natural 


Gi 


THE  PSYCHOLOGY  OF  NUMBER. 


unity,  the  limit  (the  mnj^nitude)  to  which  all  number 
refers ;  (h)  the  unit  of  measurement  (the  particular 
thing)  which  helps  to  construct  the  whole ;  and  (c)  the 
process  of  measuring,  by  which  the  second  of  these 
factors  is  used  to  make  up  or  define  the  first — thus  de- 
terminins:  its  numerical  value. 

(a)  Only  this  method  ]iresents  naturally  the  idea  of 
a  magnitude  from  which  to  set  out.  The  end  to  be 
reached,  the  object  to  be  measured,  supplies  this  idea 
of  a  given  quantity,  and  thus  gives  a  natural  basis  for 
the  develo})ment  and  use  of  ideas  of  number.  In  num- 
bers simply  as  objects,  or  in  things  s'lDiply  as  observed 
things,  there  is  no  princi})le  of  unity,  no  basis  for  nat- 
ural generalization.  Only  using  the  various  things  for 
a  certain  end  brings  them  together  into  one  ;  we  count 
and  measure  some  (piantitative  irhole. 

(h)  While  every  object  is  a  whole  in  itself,  a  unity 
in  so  far  as  it  represents  one  single  act,  no  object  sim- 
ply as  an  observed  object  is  a  vn/'f.  Objects  which  'ive 
recognise  as  three  in  number  may  l)e  before  the  child's 
senses,  and  yet  there  may  be  no  consciousness  of  them 
as  three  diUerent  units,  or  of  the  sum  three.  Some 
writers  tell  us  that  each  object  is  one,  and  so  gives 
the  natural  ]){isis  for  the  evolution  of  nund)er  ;  that 
the  starting  ])oint  is  one  o])ject,  to  which  another  ob- 
ject is  *'' addt  J,"  then  a  third,  etc.  But  this  overlooks 
j  the  fact  that  each  object  is  one,  not  a  twit  but  one 
'irhifle.  differinii:  from  and  exclusive  of  every  other 
whole.  That  is,  to  take  it  as  an  (>h.^('rve(l  ol>ject  is  to 
centre  attention  wholly  u])on  the  thing  itself  ;  attention 
would  discriininate  and  unify  the  (pialities  which  make 
the  thing  what  it  is— a  q>udltative  whole  ;  but  there 


THE  ORIGIN  OF  NUMJ3ER. 


65 


would  be  little  room  for  the  abstracting  and  relating 
action  involved  in  all  nniiiber.  A  numerical  unit  is  not 
merely  a  whole,  a  unity  in  itself,  but  is,  as  we  have  seen, 
a  unity  employed  as  a  means  for  constructing  or  meas- 
uring some  larger  whole.  Only  this  iise,  theji,  traiis- 
for)ns  the  ohject  from  a  qiialitatwe  %mity  into  a  im- 
onei'ical  imit.  The  sequence  therefore  is  :  iirst  the 
vague  unity  or  wdiole,  then  discriminated  })arts,  then 
the  recognition  of  these  parts  as  measuring  the  whole, 
wliicli  is  now  a  defined  unity — a  sum.  Or,  briefly,  the 
undefined  whole  ;  the  parts  ;  the  related  parts  i^iow 
units) ;  the  sum, 

(c)  Beginning  with  the  numbers  in  themselves,  as 
represented  by  mere  symbols,  or  with  perceived  objects 
in  themselves,  there  is  no  intrinsic  reason,  710  reason  m 
t/ni  nihid  itself  for  performing  the  operations  of  put- 
ting together  ])arts  to  make  a  whole  (using  the  unit  to 
measure  the  mngnitude),  or  of  breaking  uj)  a  whole  into 
units — discovering  the  standard  of  reference  for  meas- 
uring a  given  unity.  These  operations,"^  from  eitlier  of 
these  standpoints,  are  purely  arhitraiw ;  we  mny,  if  v»e 
wish,  do  something  with  number,  or  rather  with  num- 
ber symbols :  the  operations  arc  not  something  that  we 
onust  do  from  the  very  nature  of  nund)er  itself.  From 
tlie  point  of  view  of  the  constructive  (or  psychical)  use 
of  objects,  this  is  reversed.  These  ])rocesses  are  simply 
])hases  of  the  act  of  eon  struct  nni.  IMoreover,  the  oper- 
ations of  ad<lition,  multipHcat'on,  division,  etc.,  in  the 
method  of  perceived  objects,  have  to  be  regarded  as 


*  It  will  bo  shown  in  a  l.-ifcr  ch.'iptiT  t  hat  all  riumerical  opem- 
tio.is  grow  out  of  this  fuiRlaiueiitttl  j»n>cess. 


66 


THE  PSYCHOLOGY  OF  NUMBER. 


i 

J  ^   f 


H 


2)hi/(^teal  lieaping  up,  physical  iucreusc,  p/u/ncal  par- 
tition ;  while  in  that  of  number  by  itself  they  are 
purely  mental  and  abstract.  From  the  standpoint  of 
the  psychological  use  of  the  things,  tliese  processes 
are  not  performed  npon  physical  things,  but  with  ref- 
erence to  establishing  definite  values ;  ^  while  each  pro- 
cess is  itself  concrete  and  actual.  It  is  not  something 
to  be  grasped  by  abstract  thought,  it  is  something 
done. 

Finally,  to  teach  symbols  instead  of  number  as  the 
instrument  of  measurement  is  to  cut  across  all  the  ex- 
isting activities,  whether  impulsiv^e  or  habitual.  To 
teach  number  as  a  property  of  observed  things  is  to  cut 
it  otT  from  all  other  activities.  To  teach  it  through  the 
close  adjustment  of  things  to  a  given  end  is  to  re-enforce 
it  by  all  the  deepest  activities. 

All  the  deepest  instinctive  and  acrpiired  tendencies 
are  towards  the  constant  use  of  means  to  realize  ends  ; 
this  is  the  law  of  all  action.  All  that  the  teaching  of 
/lumber  has  to  do,  when  based  upon  the  principle  of 
rationally  u^ing  things,  is  to  make  this  tendency  more 
detinite  and  accurate.  It  simply  directs  and  adjusts 
this  piocess,  so  that  we  notice  its  various  factors  and 
measure  them  in  their  relation  to  one  another.     ^AFore- 


*  'I  ..I  conipHi'utioiis  iutroducod  in  schools — o.  g.,  that  you  can 
not  multiply  l)y  u  fraction,  nor  increase  a  number  by  division,  etc., 
because  multiplication  means  increase,  etc. — result  from  conceiving 
the  operations  as  physical  ai^f^repition  or  separation  instead  of  syn- 
thesis and  analysis  of  values — mental  i)rocesses.  To  multiply  |1() 
by  one  third  is  absurd  if  multiplication  means  a  physical  increase; 
if  it  means  a  measurement  of  value,  takii'g  a  nuuK  rical  value  of 
$\i)  (a  measured  (plant it y)  in  a  certain  way  to  iind  the  resulting 
numerical  value,  it  is  perfectly  rational. 


M 


THE  ORIGIN  OF  NUMBER. 


67 


over,  it  relics  constantly  upon  the  principle  of  rhythm, 
the  regular  breaking  up  and  putting  together  of  minor 
activities  into  a  whole ;  a  natural  principle,  and  the  basis 
of  all  easy,  graceful,  and  satisfactory  activity. 


i     >i 


'      I 


CIIAPTEE  y. 

THE   DEFINITION,    ASPECTS,    ANJ)   FACTORS   OF   NUMERICAL 

IDEAS. 

We  may  sum  up  the  steps  already  taken  as  follows : 

(1)  The  limitation  of  an  energy  (or  quality)  transforms 
it  into  quantity,  giving  it  a  certain  undelined  muchness 
or  magnitude,  as  illustrated  by  size,  hulk,  weight,  etc. 

(2)  This  indefinite  whole  of  quantity  is  transformed 
into  definite  7uimerical  value  through  the  process  of 
measurement.  (3)  This  measuring  takes  place  through 
the  use  of  units  of  magnitude,  by  putting  thetn  together 
till  they  make  '^p  an  equivalent  value.  (4)  Only  when 
this  unit  of  magnitude  has  been  itself  measured  (has 
itself  a  definite  mnierical  value)  is  the  measurement  of 
the  whole  mai^nitude  or  construction  of  the  entire  nu- 
merical  value  adequate.  Forty  feet  denotes  an  ade- 
quately measured  (piantity,  ])ecause  the  unit  is  itself 
defined  ;  f^rty  eggs  denotes  an  inade(iuately  measured 
(juantity,  Ijecause  the  unit  of  measure  is  not  d«  finite. 
Were  eggs  to  become  worth,  say,  twenty  t'nies  as  much 
as  they  are  now  worth,  they  woidd  Ik-  weighed  out  by 
the  pound — that  is,  inexact  measurement  would  give 
way  to  exact  measurement.  Having  l)efore  us,  then, 
the  psychological  process  whi^*h  c«  tistitutes  measured 
quantity,  w^e  may  define  nu!ii[*-r. 


vV.5 


DEFINITION  OP  NUMBER. 


C9 


II 


Definition  of  Xumuer. — The  simplest  expression  of 
quantity  in  numerical  terms  involves  two  components  : 

1.  A  Standard  Unit  /  a  Unit  of  lifference. — This 
is  itself  a  magnitude  necessarily  of  the  same  kind  as  tlie 
quantity  to  be  measured.  Or,  as  it  may  be  otherwise 
expressed,  the  unity  of  (piantity  to  be  measured  and 
the  unit  of  quantity  which  measures  it  are  homogeiieous 
quantities.  Thus,  inch  and  foot  (measuring  unit  and 
measured  unity),  pound  and  ton,  minute  and  hour,  dime 
and  dollar  are  pairs  of  homogeneous  (piantities. 

2.  Numerical  Value. — This  expresses  hoic  many  of 
the  standard  units  make  up,  or  construct,  the  quantity 
needing  measurement.  Examples  of  numerical  value 
are  :  the  yard  of  cloth  costs  seventeen  cents  ;  the  box 
will  hold  thirty-six  cubic  inches;  the  purse  contains 
eight  ten-dollar  pieces.  The  seventeen,  thirty-six,  eight 
represent  just  so  many  units  of  measurement,  the  cent, 
the  cubic  inch,  the  ten-dollar  piece  ;  they  express  the 
numerical  values  of  the  quantities  ;  they  are  pure  nmn- 
he/'s,  the  results  of  a  purely  mental  process.  The  nu- 
merical value  alone  represents  the  relative  value  or  ratio 
of  the  measured  quantity  to  the  unit  of  measurement. 
The  numerical  value  and  the  unit  of  measurement  taken 
together  express  the  absolute  value  (or  magnitude)  of 
tlie  measured  (piantity. 

In  the  teaching  of  aritlnnetic  much  confusion  arises 
from  the  mistake  of  identifying  numerical  value  with 
ab^ohite  magnitude — that  is,  nHtnhei\  the  instrument  of 
measurement  witli  nieasured  quantity.  Number  is  the 
product  of  the  mere  rej^etitinn  of  a  unit  of  measure- 
ment;  it  simply  indicates  hoiv  many  there  are;  it  is 
])urely  abstract,  denoting  tin  series  of  acts  by  which 


'r 


70 


THE  PSYCHOLOGY  OF   NUMBER. 


tlie  mind   constructs  defined  parts  into  a  nniiied  and 
definite  wliole.     Absolute  value  (quantity  numerically 
defined)  is  represented  by  the  application  of  this  /ww 
manij  to  magnitude,  to  quantity — that  is,  to  limited  qual- 
ity.    To  take  an  example  of  the  confusion  referred  to : 
we  are  told  that  division  is  dividing  a  (1)  number  into 
a  (2)  number  of  e(pial  (3)  numbers.     This  definition  as 
it  stands  has  absolutely  no  meaning  ;  there  is  confusion 
of   number  with   measured    (piantity.      Doubtless   the 
definition  is  intended  to  mean  :  division  is  dividing  a 
certain  definite  (juantity  into  a  number  of  definite  (pian- 
tities  equal  to  one  another.    Only  in  (2),  in  the  definition 
as  quoted,  is  the  term  number  correctly  used  ;  in  both 
(1)  and  (8)  it  means  a  measured  magnitude.     A  meas- 
nred  or  numbered  (piantity  may  be  divided  into  a  num- 
ber of  parts,  or  taken  a  number  of  times  ;  but  no  num- 
ber can  be  multiplied  or  divided  into  parts.     Number 
simpUj  as  number  always  signifies  how  many  times  one 
"so  much,"  the  unit  of  measurement,  i.s  taken  to  make 
np  another  "so  much,"  the  magnitude  to  be  measured. 
It  is,  as  already  said,  due  to  the  fundamental  activities 
of  mind,  discrimination,  and  relation,  working  upon  a 
qualitative  whole ;  and  we  might  as  well  talk  of  multi- 
plying hardness  and  redness,  or  of  dividing  them  into 
hard  and  red  things,  as  to  talk  of  multiply:?ig  a  number 
or  of  di voiding  it  into  parts. 

It  may  be  observed  that  the  problems  constantly 
used  in  our  arithmetics,  multiply  2  by  4,  divide  8  by  4, 
are  legitimate  enough  provided  they  are  properly  inter- 
preted, if  not  orally  at  least  mentally,  but  taken  literally 
are  absurd.  The  first  expression  means,  of  course,  that 
a  quantity  having  a  value  of  two  units  of  a  certain  kind 


DEFINITION  OF  NUMBER. 


n 


is  to  be  taken  four  times ;  and  similarly  8  -i-  4  means  that 
a  total  quantity  of  a  certain  kind  is  measured  by  four 
units  or  by  two  units  of  the  same  kind.  Of  course,  in 
all  mathematical  calculations  we  ultimately  operate  with 
])ure  symbols,  and  the  operations  do  not  affect  the  unit 
of  measure  ;  but  in  the  beginning  we  should  make  con- 
stant reference  to  measured  quantity,  and  always  should 
be  })repared  to  interpret  the  syml)ols  and  the  processes. 

3.  Is  umber,  then,  as  distinct  from  the  magnitude 
which  is  the  unit  of  reference,  and  from  the  nuigni- 
tude  which  is  the  unity  or  limited  quality  to  be  meas- 
ured, is  : 

The  repetition  of  a  certain  magnitude  used  as  the 
unit  of  measurement  to  e(|ual  or  express  the  compara- 
tive value  of  a  magnitude  of  the  same  kind.  It  always 
answers  the  question  ''How  many  ?" 

This  "  how  many  "  may  assume  tw  o  related  aspects : 
either  how  many  tunes  one  pai't  as  unit  has  to  be  taken 
or  repeated  to  make  up  the  whole  quantity  ;  or  how 
many  parts  as  units,  each  taken  once,  compose  the 
whole.  In  the  first  case,  the  times  of  repetition  of  the 
measuring  unit  is  mentally  the  more  prominent ;  in  the 
second,  the  actual  number  of  measuring  parts  ;  e.  g.,  in 
thinking  of  forty  yards,  we  may  at  one  time  dwell  on 
the  forty  t tines  the  unit  is  repeated  ;  at  another  time, 
on  the  actual  forty  parts  making  the  uniiied  whole. 

As  already  said,  the  number  and  the  measuring  unit 
together  give  the  absolute  magnitude  of  the  quantity. 
The  number  bv  itself  indicates  its  relative  value.  It 
always  expresses  ratio* — i.  e.,  the  relation  which  the 

*  llcnct\  again,  tlie  absurdity  of  iniiUiplying  pure  iiutnhor  or 
dividing  it  into  parts.     Wo  may  divide  a  ratio,  but  not  into  parts. 


s,* 


T2 


THE  PSYCHOLOGY  OF  NUiMRKR. 


ii  < 


r  fi 


If! 


i 


magnitude  to  bo  measured  bears  to  the  unit  of  refer- 
ence. Seven,  as  i)ure  number,  expi'esses  ecjually  tlie 
ratio  of  1  foot  to  7  feet,  of  1  inch  to  7  inches,  of  1  day 
to  1  week,  of  $1,000  to  $7,000,  and  so  on  indelinitely. 
Simply  as  seven  it  has  no  meaning,  no  definite  vahie  at 
all ;  it  only  states  a  possible  measurement. 

Tliis  definition  arrived  at  from  psychological  analy- 
sis is  that  given  by  some  of  the  greatest  mathematicians 
on  a  strictly  mathematical  basis,  as  may  be  seen  from 
comparison  with  the  following  definitions  : 

Wewtoiis. — Number  is  the  abstract  ratio  of  one 
quantity  to  another  quantity  of  the  same  kind. 

Uulers. — ]N"unil)er  is  the  ratio  of  one  quantity  to 
another  (pumtity  taken  as  unit.'^ 

Phases  of  Xumuek. — The  aspects  of  number  follow 
directly  from  what  has  been  said.  Quantity,  the  unity 
measured,  whether  a  ''collection  of  objects"  or  a  l)hys- 
ical  whole,  is  c())dinuous.  an  undefined  how  much  j  num- 
ber as  measuring  value  is  discrete,  how  many.  The  mag- 
nitude, muchness,  hefore  measurement  is  mere  unity  ; 
cfftiP  njeasurement  it  is  a  sum  taken  as  an  integer — that 
is,  an  aggregation  of  parts  (units)  making  up  one  whole; 
number  as  showing  how  many  refers  to  the  units,  which 
put  together  make  the  sum.  Quantity,  measured  mag- 
nitude, is  always  concrete  ;  it  is  a  certain  kind  of  mag- 
nitude, length,  volume,  weight,  area,  amount  of  cost. 


*  J.  C.  Gliishan,  one  of  the  ticutcst  of  living  mathenuiticians,  de- 
fines thus  :  "  A  unit  is  any  standard  of  reference  employed  in  count- 
ing any  collection  of  objects,  or  in  measuring  any  magnitude.  A 
number  is  that  which  is  a])plied  to  a  unit  to  exj)ress  the  comparative 
magnitude  of  a  (juantity  of  the  same  kind  as  the  unit."  (See  his  Arith- 
metic far  High  Schools,  etc.) 


DEFINITION   OP  NUMBER. 


etc.      "  Number,"  as   sim])ly  detiniiig   the   Low  many 
units  of  measurement,  is  always  abstract. 

The  conception  of  measuring ^>a/'^6'  and  of  times  of 
repetition  is  inseparable  from  number  as  expressing  the 
numerical  value  of  a  (quantity ;  as  discrete,  it  is  so  many  ■ 
parts  taken  one  time — constituting  the  unity  ;  as  ab- 
stract, it  is  one  part  taken  so  many  times.  In  the  one 
ease,  as  before  suggested,  attention  is  more  upon  the 
numbered ^y'tt/'^*',  in  the  other,  more  upon  the  nunther  of 
the  parts.  They  are  absolutely  correlative  conceptions 
of  the  same  measured  magnitude.  That  is,  a  value  of 
$50  may  be  regarded  as  determined  by  taking  ^IJif'tf/ 
times,  or  by  taking  $50 — that  is,  a  w/io/e  of  Jiftf/  jjarts 
— 07ie  time.  The  numerical  process  and  the  resulting 
numerical  value  are  the  same,  however  we  arrive  at  the 
numhcy- — i.  e.,  the  ratio  of  measured  quantity  to  measur- 
ing unit.  As  this  conception  of  the  relation  between 
parts  and  times  in  the  measurement  of  quantity  is  essen- 
tial to  the  interpretation  of  numerical  operations,  we 
may  give  it  a  little  further  consideration. 

AVe  wish  to  know  the  amount  of  money  in  a  roll  of 
dollar  bills.  We  take  five  dollars,  say,  as  a  convenient 
measuring  unit ;  we  separate  our  undefined  whole  into 
grou])s  of  live  dollars  each  ;  we  count  these  groups  and 
find  that  there  are  ten  of  them — i.  e.,  the  numerical 
value  is  ten  ;  we  have  now  a  definite  idea  both  of  the 
measuring  unit  and  of  the  times  it  is  repeated,  and  so 
have  reached  a  definite  idea  of  the  amount  of  money  in 
the  roll  of  bills.  "We  began  with  a  vague  whole,  an  un- 
defined unity  ;  we  broke  it  up  into  parts  (analysis),  and 
by  relating  (counting)  the  parts  we  arrived  at  our  unity 
again  ;  the  same  unity,  yet  not  the  same  as  regards  the 


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THE   PSYCHOLOGY  OF  NUMBER. 


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attitude  of  the  mind  towards  it.  It  is  now  a  definite 
unity  constituted  by  a  known  number  of  definite  jmrts  ; 
it  is  a  sifht  of  vnits.  On  the  analytic  side  ot  this 
defining  process  the  emphasis  is  on  the  parts,  the 
units  ;  on  the  syntlietic  side  the  emphasis  is  on  tlie 
defined  unity,  the  sum.  The  j)arts  are  means  to  an 
end  ;  they  exist  only  for  the  sake  of  the  end,  the 
sum.  The  ten  units — that  is,  the  unit  repeated  ten 
times — make  up,  are,  the  one  sum — i.  e.,  the  sum  taken 
one  time. 

Further,  since  the  unit  of  measure  is  itself  measured 
by  a  smaller  unit,  the  dollar,  the  same  psychological 
explanation  applies  to  the  measurement  of  the  quantity 
by  means  of  this  smaller  unit.  The  five-dollar  unit  taken 
ten  times  is  identical  with  ten  of  these  units  taken  once. 
We  are  conscious,  also,  that  any  part  of  this  iive-dollar 
unit  taken  ten  tiines  is  identical  with  ten  such  parts  taken 
once.  That  is,  $1,  taken  ten  times,  is  a  whole  of  $10 
taken  once ;  and  since  this  is  true  of  every  dollar  in  the 
five,  our  measurement  gives  a  whole  of  $10  taken  once,  a 
whole  of  $10  taken  twice,  and  so  on ;  that  is,  altogether, 
a  whole  of  $10  taken  Jive  times.  In  other  words,  the 
measurement,  ten  groups  (or  units)  of  five  dollars  each 
necessarily  implies  the  correlative  measurement.  Jive 
groups  of  ten  dollars  each. 

This  rhythmic  process  of  parting  and  wholing  which 
leads  to  all  definite  quantitative  ideas,  and  involves  the 
correlation  of  times  and  parts,  may  be  illustrated  by 
simple  intuitions.  In  measuring  a  certain  length  we 
find  it,  let  us  suppose,  to  contain  four  parts  of  three 
feet  each ;  then  the  relation  between  parts  (measuring 
units)  and  numerical  value  (times  of  repetition)  may  be 


TIMES  AND  PARTS. 


75 


perceived  in  the  following,  where  the  dots  symbolize 
both  times  and  units  of  quantity : 


•     •     • 


a 
b 


Measuring  l)y  the  3-feet  unit  we  count  it  off  four  times 
— that  is,  the  (juantity  is  expressed  l)y  3  feet  taken  four 
times.  This  is  represented  by  the  four  vertical  columns 
of  three  minor  units  each.  J]ut  this  measuring  process 
necessarily  involves  the  correlated  process  which  is  ex- 
pressed by  4  feet  taken  three  times.  For,  in  measuring 
by  three  feet,  and  finding  that  it  is  repeated  four  times, 
we  perceive  that  each  of  its  tln-ee  parts  is  repeated  /"our 
times,  giving  the  three  horizontal  rows  a,  h,  c — that  is 
to  say,  a  is  one  whole  of  4  feet,  h  a  second  whole  of 
4  feet,  and  c  a  third  whole  of  4  feet ;  or,  in  all,  4  feet 
taken  three  times.  Briefly,  1  foot  four  times  is  one 
wliole  of  4  feet ;  this  is  true  of  every  foot  of  the  origi- 
nal measure,  3  feet;  and  therefore  3  iiiGt  four  times  is 
4  feet  three  times. 

It  is  clear  that  the  two  questions,  {a)  in  12  feet  how 
many  counts  of  4  feet  each,  and  {h)  how  many  feet  in 
each  of  4  counts  making  12  feet,  are  solved  in  e^ractJij 
the  same  XLHiy  •  neitlier  the  three  counts  (times)  in  the 
first  case  nor  the  three  feet  in  the  second  case  can  be 
found  irlthout  eounthuj  the  twelve  feet  off  in  (jroujKs  (f  • 
four  feet  each. 

This  necessary  correlation,  in  the  measurement  of 
quantity,  between  "parts"  and  "times" — numerical 
value  of  the  measurincr  unit  and  numerical  value  of  the 
measured  quantity — gives  the  psychology  of  the  fun- 


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V'5l 


76 


THE  PSYCHOLOGY  OF  NUMBER. 


fl 


damenttal  principle  in  multiplication  known  as  the  law 
of  eonmuitation  :  the  product  of  factors  is  the  same  in 
whatever  order  they  may  be  taken — i.  e.,  in  the  case  of 
two  factors,  for  example,  either  may  be  multiplicand  or 
nndtiplier ;  a  times  h  is  identical  with  h  times  a. 

It  is  asserted  i)y  some  w^riters  that  this  connnntative 
law  does  not  hold  when  the  mukiplicand  is  concrete ; 
''for,"'  we  are  told,  ''though  there  is  meaning  in  re- 
quiring ^i  to  be  taken  three  times,  there  is  no  sense 
in  proposing  that  the  number  3  be  taken  four-doHars 
times" — which  is  perfectly  true.  Nevertheless,  the  ob- 
jection seems  to  be  founded  on  a  misconception  of  the 
psychical  nature  of  mnnber  and  the  i^sychological  basis 
of  the  law  of  commutation.  Psychologically  speaking, 
can  the  multiplicand  ever  ha  a  pure  number?  If  the 
foreffoinir  account  of  the  nature  of  number  is  correct, 
the  multiplicand,  however  written,  nuist  always  be  nn- 
derstood  to  express  measured  quantity  ;  it  is  always 
concrete.  As  already  said,  4x8  must  mean  4  units  of 
measurement  taken  three  times.  If  number  in  itself 
is  purely  mental,  a  result  of  the  mind's  fundamental 
process  of  analysis-synthesis — what  is  the  meaning  of 
8x4  where  both  symbols  represent  pure  numbers,  and 
where,  it  is  said,  the  law  of  commutation  does  hold  ? 
There  is  no  sense,  indeed,  in  proposing  to  multiply  three 
by  four  dollars;  but  equally  meaningless  is  the  propo- 
sition to  multiply  one  pure  numl)er  by  another — to  take 
an  abstraction  a  number  of  timcp. 

Thus,  if  the  commutative  law  "  does  not  hold  when 
the  multiplicand  is  concrete'' — indicating  a  measured 
quantity — it  does  not  hold  at  all ;  there  is  no  such  law. 
Lut  if  the  psychological  explanation  of  number  as  aris- 


TIMES  AND   PARTS. 


77 


ing  from  measurement  is  true,  there  is  a  law  of  com- 
mutation. We  measure,  for  example,  a  quantity  of  20 
pounds  weight  by  a  4:-pound  weight,  and  the  result  is 
expressed  by  4  pounds  x  5,  but  the  psychological  cor^ 
relate  is  5  pounds  x  4.  Here  we  have  true  commuta- 
tion of  the  factors,  inasmuch  as  there  is  an  interchange 
of  both  character  and  function :  the  symbol  which  de- 
notes  measured  quantity  in  the  one  expression  denotes 
pure  number  in  the  other,  and  vice  verm.  If  the  4 
pounds  in  the  one  expression  remained  4  pounds  in  the 
commuted  expression,  would  there  be  commutation  ? 

We  have  referred  to  the  fallacy  of  identifying  actual 
measuring  parts  with  numerical  value ;  it  may  now  be 
said  that,  on  the  other  hand,  failure  to  note  their  neces- 
sary connection — their  law  of  commutation — is  often  a 
source  of  perplexity.  To  say  nothing  at  present  of  the 
mystery  of  "Division,"  witness  the  discussions  upon  the 
rules  for  the  reduction  of  compound  quantities  and  of 
mixed  numbers  to  fractions.  To  reduce  41  yards  to 
feet  w^e  are,  according  to  some  of  the  rules,  to  multiply 
41  by  3.  According  to  others,  this  is  wrong,  giving  123 
yards  for  product ;  and  we  ought  to  multiply  3  feet  by 
41,  thus  getting  the  true  result,  123  feet.  Some  rule- 
makers  tell  us  that  though  the  former  rule  is  wrong  it 
may  be  followed,  because  it  always  brings  the  same  nu- 
mcrical  result  as  the  correct  rule,  and  in  ])ractice  is  gen- 
erally more  convenient.  It  seems  curious  that  the  rule 
should  be  always  wronc:  vet  always  brinii:  the  riffht  re- 
suits.  AVitli  the  relation  between  parts  and  times  before 
us  the  difficulty  yanishes.  The  expression  41  yards  de- 
notes a  measured  quan'iity ;  41  expresses  the  numerical 
value  of  it,  and  one  vard  the  measuring  unit;  our  con- 


'  1 1 


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78 


THE  PSYCHOLOGY  OF  NUMBER. 


I.  .{ 


i 


11-  f. 


»i 


ception  of  the  quantity  is  therefore,  primarily,  41  parts 
of  3  feet  each,  and  we  multiply  3  feet  by  41 ;  but  this 
conception  involves  its   correlate,  3  parts  of  41   feet 
each ;  and  so,  if  it  is  more  convenient,  we  may  multi 
ply  41  feet  by  3. 

A  similar  explanation  is  applicable  to  the  reduction, 
e.  g.,  of  '^3 J  to  an  improper  fraction.  The  denominator 
of  the  fraction  indicates  what  is,  in  this  case,  the  direct 
unit  of  measure,  one  of  the  four  equal  parts  of  the 
dollar;  and  so  we  conceive  the  $3  as  denoting  3  parts 
of  4  units  (quarter  dollars)  each,  and  multiply  4  by  3; 
or,  as  denoting  4  parts  of  3  units  each,  and  multiply 
3  by  4.  / 

Educational  Applications. 

1.  Every  numerical  operation  involves  three  factors, 
and  can  be  naturally  and  completely  apprehended  only 
when  those  three  factors  are  introduced.  This  does  not 
mean  that  they  must  be  always  formulated.  On  the 
contrary,  the  formulation,  at  the  outset,  would  be  con- 
fusing ;  it  would  be  too  great  a  tax  on  attention.  But 
the  three  factors  must  be  there  and  must  be  used. 

Every  problem  and  operation  should  (1)  proceed 
upon  the  hasis  of  a  total  magnitude — a  unity  having  a 
certain  numerical  value,  should  (2)  have  a  certain  unit 
which  measures  this  whole,  and  should  (3)  have  num- 
ber— the  ratio  of  one  of  these  to  the  other.  Suppose  it 
is  a  simple  case  of  addition.  John  has  $^2,  James  $3, 
Alfred  $4:.  IIow  much  have  they  altogether  ?  (1)  The 
total  magnitude,  the  amount  (muchness)  altogether,  is 
here  the  thing  sought  for.  There  will  be  meaning  to 
the  problem,  then,  just  in  so  far  as  the  child  feels  this 
amount  altogether  as  the  whole  of  the  various  parts. 


1  ,  v> 


EDUCATIONAL  APPLICATIONS. 


79 


(2)  The  unit  of  measurement  is  the  one  dollar.  (3)  The 
number  is  tlie  measuring  of  how  many  of  these  units 
there  are  in  all,  namely,  nine.  When  discovered  it  de- 
lines  or  measures  the  how  much  of  the  magnitude  which 
at  lirst  is  but  vaguely  conceived.  In  other  words,  it 
must  be  borne  in  mind  that  the  thought  of  some  inclu- 
sive magnitude  must,  psychologically,  precede  the  oper- 
ation, if  its  real  meaning  is  to  be  apprehended.  The 
conclusion  simply  defines  or  states  exactly  how  much  is 
that  magnitude  which,  at  the  outset,  is  grasped  only 
vaguely  as  7nere  magnitude. 

Are  we  never,  then,  to  introduce  problems  dealing 
with  simple  numbers,  with  numbers  not  attached  to 
magnitude,  not  measuring  values  of  some  kind  ;  are  we 
not  to  add  4,  5,  7,  8,  etc.  \  Must  it  always  be  4  apples, 
or  dollars,  or  feet,  or  some  other  concrete  magnitude  ? 
iV/>,  not  necessarily  as  matter  of  practice  in  getting  fa- 
cility in  handling  numhers,  Number  is  the  tool  of 
measurement,  and  it  requires  considerable  practice  with 
the  tool,  as  a  tool,  to  handle  it  with  ease  and  accuracy. 
But  this  drill  or  practice-work  in  "  number"  should  never 
be  introduced  until  after  work  based  upon  definite 
magnitudes ;  it  should  be  introduced  only  as  there  is 
formed  the  mental  habit  of  continually  referring  number 
to  the  magnitude  which  it  measures.  Even  in  the  case 
of  practice,  it  would  be  safer  for  the  teacher  to  call 
attention  to  his  reference  of  number  to  concrete  values 
in  every  case  than  to  go  to  the  other  extreme,  and  neg- 
lect to  call  attention  to  its  use  in  defining  quantity. 
I'or  example,  when  adding  "numbers,"  the  teacher 
might  say,  "  Now,  this  time  we  have  piles  of  apples,  or 
we  have  inches,  etc.,  and  we  want  to  see  how  much 


I :; 


'It' 


80 


THE  PSYCHOLOGY  OF  NUMBER. 


i.:! 


I 


iiii 


we  have  in  all "  ;  or  the  teacher  might  ask,  at  the  end  of 
every  problem,  "What  were  we  counting  up  or  measur- 
ing that  time?"  letting  each  one  interpret  as  he  pleased. 
Just  how  far  this  is  carried  is  a  matter  of  detail ;  what 
is  not  a  matter  of  detail  is  that  the  habit  of  interpreta- 
tion be  formed  by  continually  referring  the  numbers  to 
some  quantity. 

2.  The  unit  is  never  to  be  taught  as  a  fixed  thing 
(e.  g.,  as  in  the  Grube  method),  but  always  as  a  unit  of 
measurement.  One  is  never  one  thing  simply,  but  al- 
ways that  one  thing  Ui<e(I  as  a  hasisfor  countmg  off  and 
thus  measuring  some  luhole  or  qttantlty.  Absolutely 
everytliing  and  anything  which  we  attend  to  is  one i  is 
made  one  by  the  very  act  of  attending.  If  we  could 
take  in  the  whole  system  of  things  in  one  observation, 
that  would  be  one ;  if  we  could  isolate  an  atom  and  look 
at  that,  it  would  equally  be  one.  The  forest  is  one  when 
we  view  it  as  a  whole ;  the  tree,  the  branch,  the  stem, 
the  leaf,  the  cell  in  the  leaf,  is  equally  one  when  it  be- 
comes the  object  or  whole  with  which  we  are  occupied. 
But  this  oneness,  this  unity  possessed  by  every  object  of 
attention,  has  nothing  but  the  name  in  common  with 
the  numerical  unit.  In  itself  it  is  not  quantitative  at 
all ;  it  is  mere  unity  of  quality,  of  meaning.  It  be- 
comes a  quantitative  unity  (a  quantity  or  magnitude) 
only  when  considered  as  limited  (page  36),  and  as  an 
end  to  be  reached  bv  the  use  of  certain  means.  It  be- 
comes  a  unit  only  when  used  as  one  of  the  means  to 
construct  a  value  equivalent  to  a  certain  other  value. 
The  assumption  that  some  one  object  is  the  natural  unit 
of  quantity,  which  is  then  increased  by  bringing  in  other 
objects,  is  the  very  opposite  of  the  truth  ;  number  does 


ii'  I 


15.1 


EDUCATIONAL  APPLICATIONS. 


81 


not  arise  at  all  until  we  cease  taking  objects  as  objects, 
and  regard  them  simply  as  parts  which  make  up  a 
whole,  as  units  whicli  measure  a  magnitude  (see  pages 
24,  42,  on  Abstraction).  It  is  perfectly  clear,  therefore, 
that  the  method  of  "  close  observation  "  of  objects  is 
essentially  vicious ;  what  is  claimed  as  its  merit  is  in 
reality  a  grave  defect.  The  child,  according  to  the  ad- 
vocates of  this  method,  ^'' sees  what  is  brought  to  his 
notice  and  sees  all  about  it."  But  in  seeing  all  about 
the  things  there  must  be  neglect  of  the  numerical  ab- 
straction which  sees  nothing  about  the  things  save  this 
alone :  they  are  parts  of  one  whole.  There  may  be  a 
discriminating  and  relating  of  qualities  which  give  the 
things  individual  meaning;  but  there  is  not  the  pro- 
cess— at  least  the  process  is  impeded — w^hich  constructs 
quantitative  units  into  a  defined  quantitative  whole. 

This  is  plainly  so  in  case  of  units  like  dollars,  inches, 
pounds,  minutes,  etc.  They  are 'units  not  in  virtue  of 
any  quality  absolutely  inherent  in  them,  but  in  virtue  of 
their  use  in  measuring  cost,  length,  w^eight,  duration, 
etc.  It  mav  be  said  that  this  is  not  so  in  the  case  of 
books,  apples,  boys,  etc. ;  that  here  each  book,  apple, 
etc.,  is  a  unit  in  itself.  But  this  is  to  fall  into  the  error 
of  separating  counting  from  measuring,  already  referred 
to.  The  book,  the  crayon,  or  the  cube,  is  a  unity,  a 
w^hole,  in  itself,  but  it  is  not  a  unit  save  as  used  to  count 
up  (value)  the  total  amount.  The  only  point  is  that 
this  counting  gives  very  crude  measurement.  The  unit, 
book,  pie,  and  so  on,  is  not  itself  measured  by  minor 
units  of  the  same  kind.  We  are  measuring  with  an 
unmeasured  unit,  and  so  the  result  of  our  measurement 
is  exceedingly  vague  and  inaccurate,  just  as  it  would  be 


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82 


THE  PSYCHOLOGY  OF  NUMBER. 


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Pi! 


to  measure  length  by  steps  which  had  themselves  no 
detinite  length  ;  cost  of  other  goods  by  potatoes  them- 
selves changing  in  value,  etc."*^ 

Further,  since  the  measuring  unit  is  itself  measured, 
is  itself  made  up  of  minor  or  sub-units,  it  may  be  of 
any  numerical  value,  denoting  two,  three,  four,  live,  six, 
etc.,  of  such  minor  units.  Twenty-five  pounds  taken  as 
a  basis  of  measurement  is  a  unit,  is  one  /  taken  with  a 
reference  to  the  minor  unit  (1  lb.),  by  which  itself  is 
measured,  it  is  a  defined  unity,  a  simi.  Tliirty-six  feet 
referred  to  a  measuring  quantity  of  three  feet  has  a 
numerical  value  twelve  ;  and  the  three  feet  is  just  as 
much  a  unit  (one,  or  1)  as  one  foot  is  in  measuring  12 
feet.  So  the  quantities  9  piles  of  silver  coin  of  ten 
dollars  each,  and  25  jiages  of  thirty  lines  each,  have 
respectively  the  numerical  values  of  9  and  25 — that  is, 
nine  units  of  measurement  ("ones")  and  twenty-five 
units  ("  ones  ")  of  measurement.  This  is  the  very  basis 
of  our  system  of  notation.  The  numerical  value,  hun- 
dred, applied  to  any  measuring  unit,  denotes  a  quantity 
consisting  of  10  ten-units  ;  the  number,  thousand,  meas- 
ures a  quantity  which  is  composed  of  10  hundred-units  ; 
the  number,  tenth,  applied  to  any  unit,  measures  that 
quantity  which  taken  ten  times  makes  up  the  unit  of 
reference ;  the  number,  hundredth,  used  with  any  meas- 
uring unit  denotes  that  quantity  which  taken  ten  times 
makes  up  one  tenth  of  the  unit  of  reference,  etc. 


i! 


*  Hence,  once  more,  the  fallacious  ideas  introduced  by  our  arith- 
metics in  illustrating  so  much  by  these  unmeasured  units  partially 
qualitative  and  only  partially  quantitative — the  pencil,  the  apple, 
the  orange,  and  the  universal  pie — and  so  little  by  the  definite 
units  of  length,  size,  weight,  money  value,  etc. 


II'     ! 


EDUCATIONAL  APPLICATIONS. 


83 


3.  The  true  method,  then,  may  be  summarised  by 
saying  that  the  proper  introduction  to  numerical  opera- 
tions is  by  presenting  the  material  in  such  a  way  as  to 
recpiire  a  mental  operation  of  rhythmic  parting  and 
wholing — that  is,  a  quality  or  magnitude  is  to  be  pre- 
sented in  such  a  way  as  to  involve  both  separation  (men- 
tal separation,  that  is,  of  values,  not  necessarily  physical 
partition)  into  parts  and  the  recom position  of  the  parts 
into  the  whole.  The  analysis  gives  possession  of  the 
unit  of  measurement;  the  synthesis,  or  recom  position, 
gives  the  absolute  value  of  the  magnitude  ;  the  process 
itself  brings  out  the  ratio,  the  pure  number. 

We  thus  see  the  fundamental  fallacy  of  the  Grube 
method  in  another  light.  Just  as,  upon  the  whole,  it 
proceeds  from  the  mere  observation  of  objects  instead 
of  from  the  constructive  use  of  them,  so  it  works  with 
fixed  units  instead  of  with  a  whole  quantity  which  is 
measured  by  the  application  of  a  unit  of  measurement. 
The  superiority  of  the  Grube  method  to  some  of  the 
other  methods,  both  in  the  way  of  introducing  objects 
instead  of  dealing  merely  with  numerical  symbols,  and 
in  the  way  of  systematic  and  definite  instead  of  hap- 
hazard and  vague  work,  has  tended  to  blind  educators 
to  its  fundamentally  bad  character,  psychologically  speak- 
ing. There  is  no  need  to  dwell  upon  this  at  length  after 
the  previous  exposition,  but  the  following  points  may 
be  noted  : 

{a)  In  proceeding  f fom  one  to  two,  then  to  three, 
etc.,  it  leaves  out  of  sight  the  principle  of  limit,  which  is 
both  mathematically  and  psychologically  fundamental. 
There  is  no  limited  quality,  no  magnitude,  with  its  own 
intrinsic  unity,  which  sets  bounds  to  and  gives  the  rea- 


i  'v 

j  \  1 

Ml 
1        j        -, 

\r 

I,; 


li 


vjtsit 


84 


THE  rSYCIIOLOGY  OF  NUMBER. 


I 


i 


1:    i 


m 


son  for  the  numerical  operation.  Number  is  separated 
from  its  reason,  its  function,  measurement  of  (juantity, 
and  so  becomes  meaningless  and  mechanical.  There 
is  no  inner  need,  no  felt  necessity,  for  performing  the 
operations  with  number.  They  are  artiiicial.  We  are 
dealing  with  parts  which  refer  to  no  whole,  with  units 
wliich  do  not  refer  to  a  magnitude.  It  is  as  sensible  as 
it  would  be  to  make  a  child  learn  all  the  various  parts 
of  a  machine,  and  carefully  conceal  from  him  the  purpose 
of  the  machine — what  it  is  for,  what  it  does — and  thus 
make  the  existence  of  the  parts  wholly  unintelligible. 

(h)  In  beginning  with  the  fixed  unit  one  object  (1), 
then  going  on  to  two  objects,  three  objects,  then  other 
fixed  units,  there  is  no  intrinsic  psychological  connection 
among  the  various  operations.  We  mai/  add,  we  maj/ 
subtract,  we  7nai/  find  a  ratio  ;  but  addition,  subtraction, 
ratio,  remain  (psychologically)  separate  processes.  Ac- 
cording to  true  psychology,  we  begin  with  a  whole  of 
quantity,  which  on  one  side  is  analysed  into  its  units  of 
measurement,  while  on  the  other  these  units  are  syn- 
thesised  to  constitute  the  value  of  the  original  magni- 
tude ;  we  have  parts  which  refer  to  a  whole,  and  units 
which  make  a  sum.  Here  the  addition  and  subtraction 
are  psychological  counterparts  ;  we  actually  perform 
both  these  operations,  whether  we  consciously  note  more 
than  one  of  them  or  not.  Similarly,  we  go  through  a 
process  of  ratio-ing  in  the  rhythmic  construction  of  the 
whole  (much)  out  of  the  units  (many) ;  the  conscious 
grasp  of  the  principle  of  ratio  will  therefore  involve  no 
new  operation,  but  simply  reflection  upon  what  we  have 
already  done.  First  one  process,  then  another,  then 
another,  and  so  on,  is  the  law  of  the  Grube  method — 


EDUCATIONAL  APPLICATIONS. 


85 


this,  in  8])ite  of  its  maxim  to  teach  all  processes  siiiiiil- 
taiieously  :*  first,  a  process  involviiiij;  all  the  numerical 
operations,  then,  as  the  power  of  attention  and  inter- 
pretation ripens,  making  the  process  already  performed 
an  ohject  of  attention  to  bring  out  what  is  involved,  is 
the  psychological  law. 

4.  The  method  which  neglects  to  recognise  number 
as  measurement  (or  definition  of  the  numerical  value  of 
a  given  magnitude),  and  considers  it  simply  as  a  plural- 
ity of  fixed  units,  necessarily  leads  to  exhausting  and 
meaningless  mechanical  drill.  The  psychological  ac- 
count shows  that  the  natural  beginning  of  number  is 
a  whole  needing  measurement ;  the  Grube  method  (with 
many  other  methods  in  all  but  name  identical  with  the 
Grube)  says  that  some  one  thing  is  the  natural  begin- 
ning from  which  we  proceed  to  two  things,  then  to 
three  things,  and  so  on.  Two,  three,  etc.,  being  fixed, 
it  becomes  necessary  to  master  each  before  going  on  to 
the  next.  Unless  four  is  exhaustively  mastered,  live 
can  not  be  understood.  The  conclusion  that  six  months 
or  a  year  should  be  spent  in  studying  numbers  from  1 
to  5,  or  from  1  to  10,  the  learner  exhausting  all  the 
combinations  in  each  lower  number  before  proceeding 
to  the  higher,  follows  quite  logically  from  the  premises. 
Yet  no  one  can  deny  that,  however  much  it  is  sought 
to  add  interest  to  this  study  (by  the  introduction  of 
various  objects,  counting  eyes,  ears,  etc.,  dividing  the 
children  into  groups,  etc.),  the  process  is  essentially  one 


*  Which,  of  course,  it  never  does.  It  only  teaches  all  of  them 
about  one  "number"  before  it  goes  on  to  another,  each  number 
being  an  entity  in  itself — which  it  ought  not  to  do.  This  matter 
of  the  various  operations  is  discussed  in  the  next  chapter. 


i^   i 


il'l 


■ -i 


i 


'      til  J 


m 


86 


THE  PSYCHOLOGY  OF  NUMBER. 


Si  i 


ill 


h 


of  mechanical  drill.  The  interest  afforded  by  the  ob- 
jects remains,  after  all,  external  and  adventitious  to 
the  numbers  themselves.  In  the  number,  as  number, 
there  is  no  variety,  but  simply  the  ever-recurring  mo- 
notony of  ringing  the  changes  on  one  and  two  and 
three,  etc.  Moreover,  the  appeal  is  constantly  made 
simply  to  the  memorising  power.  These  combinations 
are  facts  to  be  learned.  All  the  emphasis  is  laid  upon 
the  products,  upon  the  accumulation  of  the  information 
that  2  plus  2  equal  4,  2  x  3  .+  1  =  7,  If  x  5  =  7,  etc. ; 
as  a  result,  the  ''numbers"  remain  something  external 
to  the  mind's  own  activity ;  something  impressed  upon 
it,  and  carried  by  it,  not  something  growing  out  of  its 
own  action  and  coming  to  be  a  normal  habit  of  intrinsic 
mental  -working. 

Contrast  with  the  True  Method. — For  the  sake  of 
indicating  more  clearly  the  defects  of  this  method,  let  us 
follow  out  the  contrast  with  the  true  or  psychological 
method : 

(a)  The  emphasis  is  all  the  time  npon  the  perform- 
ance of  a  certain  mental  process  ;  the  product,  the  par- 
ticular fact  or  item  of  information  to  be  grasped,  is  sim- 
ply the  outcome  of  this  process.  There  is  a  given  whole 
to  be  counted  off  into  minor  wholes ;  a  group  of  objects 
to  be  marked  off  into  sub-gronps ;  a  given  magnitude 
of  surface  to  be  cut  up  into  equal  minor  units  of  sur- 
face ;  a  weight  to  be  measured  through  equalising  it 
with  a  number  of  sub-units  of  weight,  etc.  Then  the 
iiumbei"  of  sub-groups,  minor  unities  or  parts,  has  to  be 
counted  up  in  order  to  find  the  numerical  value  of  the 
original  whole.  The  entire  interest  is  in  the  actual 
process  of  distinguishing  the  whole  into  its  parts,  and 


EDUCATIONAL  APPLICATIONS. 


87 


combining  the  parts  so  as  to  make  up  the  vahie  of  the 
whole.  Wherever  there  is  a  break  with  the  mind's  own 
activity,  there  the  facts  or  principles  learned  are  exter- 
nal, and  interest  must  be  partial  and  defective.  The 
operation  becomes  mechanical,  and  the  operator  a  mere 
machine ;  or  else  it  is  maintained  only  by  a  series  of 
artilicial  stimulations,  which  keep  the  mind  in  a  con- 
dition of  strain — an  effort  which  has  its  sole  source  in 
the  need  of  covering  the  gap  between  the  intrinsic  men- 
tal activity  and  the  abnormal  action  which  is  forced 
upon  the  mind.  Wherever  there  is  intrinsic  mental 
activity  there  is  interest ;  interest  is  nothing  but  the 
consciousness  arising  from  normal  activity.*  Besides, 
this  a^'tivity  of  parting  and  wholing,  of  measuring  off 
into  minor  units  of  value,  and  summing  up  these  minor 
units  into  the  one  whole,  can  not  be  performed  without 
the  mind's  getting  the  information  needed,  that,  e.  g., 
l+l-f-l  +  l,  or  1  +  1  +  2,  etc.,  =  4. 

{h)  The  appeal,  according  to  the  psychological  method 
(number  as  mode  of  measurement),  is  not  to  memory  or 
memorising,  but  is  a  training  of  attention  and  judgment ; 
and  this  training,  which  forms  the  halit  of  definite  analy- 
sis and  synthesis,  forms  the  habit  of  the  rhythmic  balanc- 
ing of  parts  against  one  another  in  a  whole,  and  the  habit 
of  the  rhythmic  or  orderly  breaking  up  of  a  whole  into 
its  definite  parts.     So  far  as  this  habit  is  formed  the 

*  Wherever  we  have  to  appeal  to  external  stimulus  to  make  a 
subject  Interesting,  it  indicates,  of  course,  that  the  activity  if  left  to 
itself  would  cease,  that  the  mind  would  wander  or  become  listless. 
This  means  that  there  is  no  intrinsic  interest,  no  spontaneous  move- 
ment, no  self-developing  energy  in  the  mind.  Wherever  there  is 
this  intrinsic  activity,  the  subject  is  interesting  of  necessity  and 
does  not  have  to  be  made  so. 


I  - 


!M! 


ii 


i  I 


i  ii 


« 


• '  % 


88 


THE  PSYCHOLOGY  OF  NUMBER. 


I>   1 


I'M 

.;  i 

i 


11    :    ) 


i 


r  I 


memory  will  take  care  of  itself.  Tlie  facts  do  not  need 
to  be  seized  and  carried  by  sheer  effort  of  memory,  but 
are  reproduced,  whenever  needed,  out  of  tlie  mind's  own 
power.  The  learning  of  facts,  the  preservation  and  re- 
teution  of  information,  is  an  outcome  of  the  formation 
of  habit,  of  the  attainment  of  power.  The  method  which 
neglects  the  measuring  function  of  number  can  not  pos- 
sibly lead  to  a  definite  habit ;  it  can  result  only  in  the 
ability  to  remember. 

There  is  no  question  here  about  the  need  of  drill,  of 
discipline,  in  all  instruction.  But  there  is  every  ques- 
tion about  the  true  nature  of  drill,  of  discipline.  The 
sole  conception  of  drill  and  of  discipline  which  can  be 
afforded  by  the  rigid  unit  method  is  that  of  ability  to 
hold  the  mind  fixed  upon  something  external,  and  of 
ability  to  carry  facts  by  sheer  force  of  memory.  By 
the  psychological  method  of  treating  the  unit  as  means 
to  an  end,  a  basis  of  measurement,  the  discipline  con- 
sists in  the  orderly  and  effective  direction  of  power 
already  struggling  for  eocpression  or  utterance.  One 
is  the  drill  of  a  slave  to  fit  him  for  a  task  which  he 
himself  does  not  understand,  and  which  he  docs  not 
care  for  in  itself.  The  other  is  the  discipline  of  the 
free  man  in  fitting  him  to  be  an  efiicient  agent  in  the 
realization  of  his  own  aims. 

{c)  Finally,  the  fixed  unit  method  deadens  interest 
and  mechanizes  the  mind  in  not  allowing  free  play  to 
its  tendencies  to  variety,  to  continual  new  development. 
As  already  said,  according  to  the  Grube  method,  the 
fact  that  2  -h  2  =  4  always  remains  precisely  the  same, 
no  matter  how  much  its  monotony  is  disguised  by  per- 
mutations with  blocks,  slioe  pegs,  pictures  of  birds,  etc. 


EDUCATIONAL  APPLICATIONS. 


89 


According  to  the  measuring  method,  the  habit  or  gen- 
eral direction  of  action  remains  the  same,  but  is  con- 
stantly differentiated  through  application  to  new  facts. 
According  to  the  Grube  method  unity  is  one  thing, 
and  that  is  the  end  of  it.  According  to  the  measuring 
method  unity  may  be  12  (the  dozen  oranges  as  meas- 
ured by  the  particular  orange,  the  day  as  measured  by 
tlie  hour,  the  foot  as  measured  by  the  inch,  the  year  as 
measured  by  the  month,  etc.),  or  it  may  be  100 — e.  g., 
the  dollar  as  measured  by  the  cent. 

Instead  of  relying  upon  a  minute  and  exhaustive 
drill  in  numbers  from  1  to  5,  allowing  next  to  no  spon- 
taneity, severing  nearly  all  connection  with  the  child's 
actual  experience,  ruling  out  all  variety  as  diametrically 
opposed  to  its  method,  it  can  lay  hold  of  and  give  free 
play  to  any  and  every  interest  in  a  whole  which  comes 
up  in  the  child's  life.  Unity  as  12,  as  a  dozen,  is  likely 
to  be  indefinitely  more  familiar  and  interesting  to  a  child 
than  7 ;  the  desire  to  be  able  to  tell  thne  comes  to  be  an 
internal  demand,  etc.  But  the  Grube  method  must  rule 
out  12.  Twenty -five  as  a  imity  (of  money,  the  quarter- 
dollar),  50  (as  the  half-dollar),  100  (as  the  dollar),  are 
continual  and  lively  interests  in  the  child's  own  activi- 
ties. Each  of  these  is  just  as  much  one  as  is  one  eye 
or  one  block,  and  is  arithmetically  a  very  much  better 
type  of  unit  than  the  block  by  itself,  because  it  is  capa- 
ble of  definite  measurement  or  rhythmic  analysis  into 
sub-units,  thus  involving  division,  multiplication,  frac- 
tions, etc. — operations  which  are  entirely  external  and 
irrelevant  to  the  fixed  unit. 

Some  will  probably  say,  "But  100,  or  even  12,  is 
altogether  too  complex  and  difiicult  a  number  for  a 


f[ : : 


ill 


Ji 


i:  m 


i: 


■'I 


90 


THE  PSYCHOLOGY  OF  NUMBER. 


t  1 


1^    I 


i 


cliild  to  grasp."  Yes,  if  it  is  treated  simply  as  an  ac- 
cumulation or  aggregation  of  individual  separate  fixed 
units.  Very  few  adults  can  definitely  grasp  100  in  that 
sense.  The  Grube  method,  proceeding  on  the  basis  of 
the  separate  individual  thing  as  unit,  is  quite  logical  in 
insisting  upon  exhausting  all  the  combinations  of  all  the 
lower  numbers.  No,  if  100  is  treated  as  a  natural  whole 
of  value,  needing  to  be  definitely  valued  by  being  meas- 
ured out  into  sub-units  of  value.  One  dollar  is  one^  we 
repeat,  as  much  as  one  block  or  one  pebble,  but  it  is 
also  (which  the  block  and  pebble  as  fixed  things  are 
not)  two  50's,  four  25's,  ten  lO's,  and  so  on. 

It  may  be  well  to  remind  the  reader  that  while  we 
are  dealing  here  only  with  the  theory  of  the  matter,  yet 
the  successful  dealing  with  such  magnitude  as  the  dozen 
and  the  dollar  is  not  a  matter  of  theory  alone.  Actual 
results  in  the  schoolroom  more  than  justify  all  that  is 
here  said  on  grounds  of  psychology.  During  the  six 
months  in  which  a  child  is  kept  monotonously  drilling 
upon  1  to  5  in  their  various  combinations,  he  may,  as 
proved  by  experience,  become  expert  in  the  combina- 
tion of  higher  numbers,  as,  for  example,  1  ten  to  5  tens, 
1  hundred  to  5  hundred,  etc.  If  the  action  of  the  mind 
is  judiciously  aided  by  use  of  objects  in  the  measuring 
process  which  gives  rise  to  number,  he  knows  that  4 
tens  and  2  tens  are  6  tens,  4  hundred  and  2  hundred 
aie  6  hundred,  etc.,  just  as  surely  as  he  knows  that  4 
cents  and  2  cents  are  6  cents ;  because  he  knows  that 
4  units  c  asurement  of  any  kind  and  2  units  of  the 
same  kind  are  6  units  of  the  same  kind.  Moreover, 
this  introduction  of  larger  quantities  and  larger  units 
of  measurement  saves  the  child  from  the  chilling  effects 


EDUCATIONAL  APPLICATIONS. 


91 


of  monotony,  maintains  and  even  increases  liis  interest 
in  numerical  operations  through  variety  and  novelty,  and 
through  constant  appeals  to  his  actual  experience. 

Many  a  child  who  has  never  seen  "  four  birds  sitting 
on  a  tree  and  two  more  birds  come  to  join  them,  mak- 
ing in  all  six  birds  sitting  on  the  tree,"  has  heard  of  one 
of  his  father's  cows  being  sold  for  $40  (4  tens),  and  an- 
other for  $20  (2  tens),  making  in  all  6  tens  or  $60 ;  or 
of  one  team  of  horses  being  sold  for  4  hundred  dollars, 
and  another  for  2  hundred  dollars,  in  all  6  hundred 
dollars. 

AVhen  it  is  urged  that  these  higher  numbers  are  be- 
yond the  child's  grasp,  what  is  really  meant  ?  If  the 
meaning  is  that  the  child  can  not  picture  the  hundred, 
cannot  visualis*^  it,  this  is  perfectly  true ;  but  it  is  about 
equally  trr  the  case  of  the  adult.  No  one  can  have 
a  perfect  mental  picture  of  a  hundred  units  of  quantity 
of  any  kind.  Yet  we  all  have  a  conception  of  a  hun- 
dred such  units,  and  can  work  with  this  conception  to 
perfectly  certain  and  valid  results.  St  a  child,  getting 
from  the  rational  use  of  concrete  objects  as  symbols  of 
measuring  units  the  fact  that  4  such  units  and  2  sucli 
units  are  6  such  units,  gets  a  clear  enough  working  con- 
ception for  any  units  whatever.  The  opposite  assump- 
tion proceeds  from  the  fallacy  of  tlie  fixed  unit  method, 
and  from  the  kindred  fallacy  that  to  know  a  quantity 
numerically  we  must  mentally  image  its  numerical  value ; 
grasp  in  one  act  of  attention  all  the  measuring  parts  con- 
tained in  the  quantity.  Neither  adult  nor  child,  we  re- 
peat, can  do  this.  We  can  not  visualise  a  figure  of  a 
thousand  sides — perhaps  but  few  of  us  can  "  picture  " 
one  of  even  ten  sides — but  we  nevertheless  know  the 


i 


m 


1:1 

1;    II 


'i   ^»     :, 
'1    , 


I 


^  ^!  i 

I 


92 


THE  PSYCHOLOGY  OP  NUMBER. 


« 


i 


■'  t' 


I! 


i 


figure,  have  a  definite  conception  of  it,  and  with  certain 
given  conditions  can  determine  accurately  the  proper- 
ties of  the  figure.  The  objection,  in  short,  proceeds 
from  the  fallacy  that  we  know  only  what  we  see ;  that 
only  what  is  presented  to  the  senses  or  to  the  sensuous 
imagination  is  known ;  and  that  the  ideal  and  universal, 
the  product  of  the  mind's  own  working  upon  the  mate- 
rials of  sense  perception,  is  not  knowledge  in  any  true 
sense  of  the  word. 

To  sum  up  :  One  metlijod  cramps  the  mind,  shutting 
out  spontaneity,  variety,  and  growth,  and  holding  the 
mind  down  to  the  repetition  of  a  few  facts.  The  other 
expands  the  mind,  demanding  the  repetition  of  activi- 
ties, and  taking  advantage  of  dawning  interest  in  every 
kind  of  value.  One  method  relies  upon  sheer  memo- 
rising, making  the  "  memory  "  a  mere  fact-carrier ;  the 
other  relies  upon  the  formation  of  habits  of  action  or 
definite  mental  powers,  and  secures  memory  of  facts  as 
a  product  of  spontaneous  activity.  One  method  either 
awakens  ?io  interest  and  therefore  stimulates  no  de- 
veloping activity  ;  or  else  appeals  to  such  extrinsic  in- 
terest as  the  skilful  teacher  may  be  able  to  induce  by 
continual  change  of  stimulus,  leading  to  a  varying  ac- 
tivity that  produces  no  unified  result  either  in  organis- 
ing power  or  in  retained  knowk  dge.  The  other  method, 
in  relying  on  the  mind's  own  activity  of  parting  and 
wlioling — its  natural  functions — secures  a  continual  sup- 
port and  re-enforcement  from  an  internal  interest  which 
is  at  once  the  condition  and  the  product  of  the  mind's 
vigorous  action. 


T 


h 
n 

<\ 

n 

f 
c 


1 


\)k 


'»s 


CHAPTER  YI. 

the  development  of  number  ;  or,  the  arithmetical 

operations. 

Numerical  Operations  as  External  and  as 
Intrinsic  to  Number. 

Addition,  Subtraction,  Multiplication. — As  we 
have  already  seen,  nninher  in  the  strict  sense  is  the 
measure  of  quantity.  It  definitely  measures  a  given 
quantity  by  denoting  how  many  units  of  measurement 
make  up  the  quantity.  All  immerical  operations,  there- 
fore, are  phases  of  this  process  of  measurement ;  these 
operations  are  bound  together  by  the  idea  of  measure- 
ment, and  they  differ  from  one  another  in  the  extent 
and  accuracy  with  which  they  carry  out  the  measuring 
idea. 

As  ordinarily  treated,  the  fundamental  operations — 
addition,  subtraction,  etc. — are  arithmetically  connected 
but  psychologically  separated.  Addition  seems  to  be 
one  operation  which  we  perform  with  numbers,  sub- 
traction another,  and  so  on.  This  follows  from  a  mis- 
conception of  the  nature  of  number  as  a  psychical 
process.  Wherever  one  is  regarded  as  one  thing^  two 
as  two  things^  three  as  three  things^  and  so  on,  this 
thought  of  numerical  operations  as  something  exter- 
nally performed  upon  or  done  with  existing  or  ready- 

93 


h 


\ 


f« 


(■      ■ 


Mi 


94 


THE  PSYCHOLOGY  OF  NUMBER. 


ill 


made  numbers  is  inevitable.  Number  is  a  fixed  exter- 
nal something  upon  which  we  can  operate  in  various 
ways :  it  simply  happens  that  these  various  ways  are 
addition,  subtraction,  etc. ;  they  are  not  intrinsic  in  the 
idea  of  number  itself. 

But  if  number  is  the  mode  of  measuring  magnitude 
— transforming  a  vague  idea  of  quantity  into  a  definite 
one — all  these  operations  are  internal  and  intrinsic  de- 
velopments of  number;  they  are  the  growth,  in  accu- 
racy and  definiteness,  of  its  measuring  power.  Our 
present  purpose,  then,  is  to  show  how  these  operations 
represent  the  development  of  number  as  the  mode  of 
measurement,  and  to  point  out  the  educational  bearing 
of  this  fact. 

The  Stages  of  Measurement. — We  have  already  ?een 
that  there  are  three  stages  of  measurement,  differing 
from  one  another  in  accuracy  and  definiteness.  We 
may  measure  a  quantity  (1)  by  means  of  a  unit  which 
is  not  itself  measured,  (2)  with  a  unit  which  is  itself 
measured  in  terms  of  a  unit  homogeneous  with  the 
quantity  to  be  measured,  (3)  with  a  unit  wliich  is  not 
only  defined  as  in  (2),  but  has  also  a  definite  relation  to 
some  quantity  of  a  different  kind.  If,  for  example,  we 
count  out  the  number  of  apples  in  a  peck  measure,  we 
are  using  the  first  type  of  measurement;  there  is  no 
minor  unit  homogeneous  with  the  peck  measure  by 
which  to  define  the  apple.  If  we  measure  the  number 
of  pounds  of  apples,  we  are  using  the  second  type  of 
measurement ;  each  apple  may  itself  be  measured  and 
defined  as  so  many  ounces,  and  as  therefore  capable  of 
exact  comparison  with  the  total  number  of  pounds.  In 
this  case  we  have  a  continuous  scale  of  homogeneous 


THE  ARITHMETICAL  OPERATIONS. 


95 


measuring  units — drachms,  ounces,  etc. ;  in  the  first  type 
of  measuring  we  have  not  such  a  scale. 

Finally,  the  pound  itself  may  be  defined  not  only  as 
16  ounces,  but  also  as  bearing  a  relation  to  some  other 
standard  ;  as,  e.  g.,  a  cubic  foot  of  distilled  water  at  the 
temperature  of  39'83  weighs  62J  pounds,  the  linear  foot 
itself  being  defined  as  a  definite  part  of  a  pendulum 
which,  under  given  conditions,  vibrates  seconds  in  a 
given  latitude  (see  page  46). 

The  Specific  Nximerical  Operations. — The  funda- 
mental operations,  as  already  said,  are  phases  in  the  de- 
velopment of  the  measuring  process. 

1.  We  have  seen  that  the  comparison  of  two  quan- 
tities in  order  to  select  the  one  fittest  for  a  given  end 
not  only  gives  rise  to  quantitative  ideas,  but  also  tends 
to  make  them  more  clear  and  definite.  Each  of  the 
quantities  is  at  first  a  vague  whole ;  but  one  is  longer 
or  shorter,  heavier  or  lighter,  in  a  word,  more  or  less 
than  the  other.  Here  we  have  the  germinal  idea  of 
addition  and  subtraction.  The  difference  between  the 
quantities  will  be  a  vague  muchness,  just  as  the  quan- 
tities themselves  are  vague,  and  will  become  better  de- 
fined just  as  these  become  better  defined.  This  better 
definition  arises  with  the  first  stage  of  measurement — 
that  of  the  undefined  unit.  We  begin  with  measuring 
a  collection  of  objects  by  counting  them  off,  and  this 
suggests  the  measuring  of  a  continuous  quantity  in  a 
similar  way — that  is,  by  counting  it  off  in  so  many  paces, 
hand-breadtlis,  etc.  Now,  in  the  use  of  the  inexact 
unit  there  is  given  a  more  definite  idea  of  the  quanti- 
ties and  of  the  more  or  less  which  distinguishes  them, 

but  no  explicit  thought  of  the  ratio  of  one  to  the  other ; 
8 


/    : 


A\'^ 


i^'^h 


M  i 


il  m 


96 


THE   PSYCHOLOGY  OF  NUMBER. 


lllj  :i 


t:i 


1.  * 


»■:' 


I  i     ■> 

:i    1 


I 


ii! 


f     i 


:)    ' 


there  is  a  counting  of  lH'e  tilings  but  not  of  ^qval  things. 
In  other  words,  the  process  of  counting  witli  an  unmeas- 
ured unit  gives  us  aritlnneticully  AdiUtio7i  and  /Suhtrac- 
tlon.  The  result  is  definite  sini})ly  as  to  mare  or  leas 
of  magnitude.  It  shows  how  many  more  coins  there 
are  in  one  heap  than  in  another,  liow  many  more  pa/;es 
in  one  distance  than  another,  in,  etc.  It  gives  an  idea  of 
the  relative  value  in  this  one  imint  of  moreness  or  aggre- 
gation^ but  it  does  not  bring  into  consciousness  what  mul- 
tiple, or  part,  one  of  the  quantities  is  of  the  other,  or  of 
their  difference.  This  is  a  more  complex  conception, 
and  so  a  later  mental  product. 

2.  With  the  development  of  the  idea  of  quantity  in 
fulness  and  accuracy  the  second  stage  of  measurement 
is  reached,  in  which  the  measuring  unit  is  uniform 
and  defined  in  terms  homogeneous  with  the  measured 
quantity. 

This  principle  of  measuring  with  an  exact  unit — 
i.  e.,  a  unit  which  is  itself  made  up  of  minor  units  in 
the  same  scale — gives  rise  to  Midtiplication  and  Divi- 
sion, and  is  in  reality  the  principle  of  ratio.  In  the 
addition  or  subtraction  of  two  quantities  we  are  not 
conscious  of  their  ratio ;  we  do  not  even  use  the  idea  of 
their  ratio.  In  multiplication  and  division  we  are  con- 
stantly dealing  with  ratio.  We  do  not  discover  merely 
that  one  quantity  is  more  or  less  than  another,  but  that 
one  is  a  certain  part  or  multiple  of  another.  When, 
for  example,  we  multiply  $4  by  5  we  are  using  ratio ; 
we  have  a  sum  of  money  measured  by  5  units  of  $4 
each,  where  the  number  5  is  the  ratio  of  the  quantity 
measured  to  the  measuring  unit.  In  division,  the  in- 
verse of  multiplication,  ratio  is  still  more  prominent. 


ri 


TT 


THE  ARITHMETICAL  OPERATIONS. 


©T 


The  idea  of  ratio  involved  in  multiplication  and 
division  is  a  much  more  practical  one  than  that  of 
mere  aggregation  (more  or  less)  involved  in  addition 
and  subtraction,  because  it  helps  to  a  more  accurate 
adjustment  of  means  to  end.  Suppose  a  man  in  re- 
ceipt of  a  certain  salary  knew  that  the  rent  of  one  of 
two  houses  is  $100  a  year  more  than  that  of  the  other, 
but  could  not  tell  the  ratio  of  the  $100  to  his  salary,  it 
is  obvious  that  he  would  have  but  little  to  guide  him  to 
a  decision.  But  if  he  knows  that  $100  is  one  fifth  or 
one  fiftieth  of  his  entire  income,  he  has  clear  and  posi- 
tive knowledge  for  his  guidance. 

"With  ratio — nmltiplication  and  division — go  the 
simpler  forms  and  processes  of  fractions.* 

3.  The  principle  of  measuring  one  scale  in  terms 
of  another  gives  us  arithmetically  ^/•^o/'^ton,  and  the 
operations  involving  it,  such  as  percentage  and  multi- 
plication and  division  of  fiactions,  and  brings  out  the 
idea  of  the  equation. 

The  Order  of  Arithmetical  Instruction. — We 
have  already  seen  one  fundamental  objection  to  the 
ordinary  method  of  teaching  number,  whether  as  car- 
ried on  in  a  haphazard  way  or  by  what  is  known  as 
"  the  Grube "  method  ;  it  takes  number  to  be  a  fixed 

*  In  external  form,  but  not  in  internal  meaning,  other  fractions 
belong  here  also.  For  example,  the  ratio  of  14  to  3  may  be  written 
14  -^  3  or  "i/ ;  in  any  case,  the  idea  is  to  discover  how  many  units  of 
the  value  of  3  measure  the  value  of  14  units,  but  the  very  fact  that 
3  is  taken  as  the  unit  shows  the  meaning  to  be  the  discovery  of  the 
ratio  of  14  to  3  as  unity.  Whether  the  result  can  actually  be  writ- 
ten in  integral  form  or  not  is  of  no  consequence  in  principle,  so  long 
as  the  process  is  the  attempt  to  discover  the  ratio  to  unity ;  the  pro- 
cess is  ^  of  14. 


1, 


I , 


ii  ii'  i 


jj 

'*  l\  ^ 

'  'i 

'^1 

1 
1 

lyjj 

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1 

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m 

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lyi 

;!  M 


11.; 


l!:i| 


! 


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.1  i 


]\i 


« > 
i 


98 


THE  rSYCIIOLOGY  OP  NUMBER. 


quantity,  instead  of  a  mental  operation  concerned  in 
measuring  quantity.  We  can  now  appreciate  another 
fundamental  objection  :  it  attempts  to  teach  all  the 
operations  simultaneously^  and  thus  neglects  the  fact 
of  growth  in  psychological  complexity  corresponding 
to  the  development  of  the  stages  of  measurement.  It 
takes  each  number  as  an  entity  in  itself,  and  exhausts 
all  the  operations  (except  formal  proportion  *)  that  can 
be  performed  within  the  range  of  that  number.  It  as- 
sumes that  the  logical  order  is  the  order  of  growth  in 
psychological  difficulty.  All  operations  are  implied 
even  in  counting^  hut  are  not  therefore  identical. 

Logically^  or  as  processes,  all  operations  are  implied^ 
even  in  counting.  To  count  up  a  total  of  four  apples 
involves  multiplication  and  division,  and  thus  ratio  and 
fractions.  When  we  have  counted  3  of  the  4  apples, 
we  have  taken  a  first  1,  a  second  1,  and  a  third  1 — 
that  is,  a  total  of  three  I's — out  of  the  4  which  com- 
pose the  original  quantity.  We  \\2iYQ  divided  the  origi- 
nal quantity  of  apples  into  partial  values  as  units,  and 
have  taken  one  of  those  nnits  so  many  times  ;  this  is 
multiplication.  But  it  does  not  follow  that,  because 
the  operations  are  logically  implied  in  this  process,  they 
are  therefore  the  same  in  their  complete  development 
and  all  equal  in  point  of  psychological  difficulty ;  much 
less  that  they  should  be  definitely  evolved  in  conscious- 
id  all  taught  together.  The  acorn  implies  the 
it  the  oak  is  not  the  acorn.    Multiplication  is  im- 


ness 


.1  i 


k 


*  Why  not  proportion,  or  even  logarithms,  on  the  principle  that 
everything  that  is  logically  correlative  should  be  taught  at  once  f 
The  logarithm  is  just  as  much  involved  in  say  8,  as  are  all  the  mul- 
tiplications and  additions  which  can  be  deduced  from  it. 


TUE  ARITHMETICAL  OPEUATIONS. 


99 


plied  in  tlie  simple  act  of  counting:,  and  lios  its  genesis 
in  addition  ;  but  multiplication  is  not  merely  counting, 
nor  is  it  identical  with  addition.  The  operation  indi- 
cated in  $2  +  82  +  $12  +  $2  =  $S  may  be  performed, 
and  in  the  initial  stages  of  mental  growth  is  performed, 
without  the  conscious  recognition  that  eight  is  four 
times  two.  The  latter  is  implied  in  the  former,  and  in 
due  tinie  is  evolved  from  it ;  but  for  this  very  reason  it 
is  a  later  and  more  complex  conception,  and  therefore 
makes  a  severer  demand  upon  conscious  attention.  The 
summing  process  is  made  comparatively  easy  through 
the  use  of  objects  ;  it  is  little  more  than  the  perception 
of  related  things.  The  nuiltiplication  process  is  more 
complex,  because  it  demands  the  actual  use  and  more 
or  less  conscious  grasp  of  ratio,  or  times,  the  abstract  ele- 
ment of  all  numbers ;  it  is  the  conception  of  the  rela- 
tion of  things.  We  might  go  on  adding  twos,  or  threes, 
or  fours,  instantly  merging  each  successive  addend  in 
the  growing  aggregate,  and,  never  returning  to  the  ad- 
dends, correctly  obtain  the  respective  sums  without  the 
more  abstract  conception  of  times  ever  arising — that  is, 
without  ever  being  conscious  that  the  "  sum  • '  is  a  prod- 
uct of  which  the  times  of  repetition  of  the  addend  is 
one  of  the  factors.  Certainly  this  more  abstract  notion 
does  not  arise  at  first  in  the  development  of  numerical 
ideas  in  either  the  child  or  the  race. 

If  anyone  still  maintains  that  addition  (of  equal  ad- 
dends) and  multiplication  are  identical  processes,  let 
him  prove  by  mere  summing  (or  counting)  that  the 
square  root  of  two,  multiplied  by  the  square  root  of 
three,  is  equal  to  the  square  root  of  six ;  or  find  by  loga- 
rithms the  sum  of  a  given  number  of  equal  addends. 


.'  fill 


ti 


i^V,:: 


III 


"I'M 


100 


THE  PSYCHOLOGY  OF  NUMBER. 


\ 


ill- 


ii  i 


•!)-^ 
'*!- 


;ii 


Simultaneoiis  Method  not  Psychological. — It  seems 
clear,  therefore,  that  the  fundamental  operations  as  for- 
mal processes  should  not  be  all  taught  together  ;  on  the 
other  hand,  rational  use  should  be  made  of  their  logical 
and  psychological  correlation.  It  is  one  thing  to  per- 
form arithematical  operations  in  such  a  way  as  to  in- 
volve the  use  of  correlative  operations,  and  it  is  another 
thing  to  force  these  operations  into  consciousness,  or 
to  make  them  the  express  object  of  attention.  The 
natural  psychological  law  in  all  cases  is  first  the  use  of 
the  process  in  a  rational  way,  and  then,  after  it  has  be- 
come familial',  abstract  recognition  of  it. 

The  method  usually  followed  violates  both  sides  of 
the  true  psychological  principle.  Because  it  treats  num- 
ber as  so  many  independent  things  or  unities,  it  can  not 
mentally  or  by  interpretation  bring  out  how  the  opera- 
tions are  correlative  with  one  another.  It  is  only  when 
the  unit  is  treated  not  as  one  thing,  but  as  a  standard 
of  measuring  numerical  values,  that  addition  and  multi- 
plication, division  and  fractions,  are  rationally  correla- 
tive. And  it  is  because  this  correlation  is  not  brought 
out  and  rationally  used  that — in  spite  of  the  teaching 
of  all  the  operations  contemporaneously — division  is 
still  a  mystery  and  fractions  a  dark  enigma. 

Then,  the  common  method  errs  in  the  opposite  ex- 
treme by  attempting  to  force  the  recognition  of  ratio, 
and  fractions,  into  consciousness  before  the  mind  is 
sufficiently  mature,  or  sufficiently  exercised  in  the  use 
of  ratio,  to  grasp  its  meaning.  The  result  of  tliis  un- 
natural method  is  that  mechanical  drill  and  memoiiz- 
ing,  with  the  sure  effect  of  waning  interest  and  feeble 
thought,  is  forced  upon  the  pupil.     To  master  all  the 


1 


THE  ARITHMETICAL  OPERATIONS. 


101 


numerical  operations  contained  in  0,  7,  8,  and  9  is  a 
slow  and  tedious  process,  and  so  the  method  is  com- 
pelled in  self-consistency  to  limit  the  range  of  numbers 
which  are  to  be  mastered  in  a  given  time.  In  reality  it 
is  easy  for  the  mind  to  grasp  the  fact  that  $1  is  a  hun- 
dred ones,  or  fifty  twos,  or  ten  tens,  or  five  twenties, 
long  before  it  has  exhausted  all  possible  operations  with 
such  numbers  as  7  or  11  or  18.  It  might,  indeed,  be 
maintained  that  a  return  to  the  old-fashioned  ways  of 
our  boyhood,  by  which  we  soon  became  expert  in  the 
mechanical  processes  of  addition  and  subtraction,  would 
be  preferable  to  this  monotonous  drill  on  "  all  that  can 
be  done  with  the  numbers"  from  1  to  10  and  from  10 
to  20  in  the  second  year ;  for  this  new  method  is  just 
about  as  mechanical  as  the  old,  and,  while  leaving  the 
child  little  if  any  better  prepared  for  the  "  analysis  "  of 
the  higher  numbers,  leaves  him  also  without  the  expert- 
ness  in  the  operations  which  is  essential  to  progress  in 
arithmetic. 

Dimsion  and  Siihtraction  not  to  precede  MuUvpli- 
catlon  and  Addition. — On  the  ground  that  the  "  first 
])rocedure  of  the  mind  is  always  analytic,"  *  some  main- 
tain that  division  and  subtraction  (the  "analytic  pro- 
cesses ")  should  be  taught  before  multiplication  and  ad- 
dition. 

But  just  as  multiplication,  definitely  using  the  idea 
of  ratio,  is  a  more  complex  process  than  addition,  so 
division,  the  inverse  of  multiplication,  is  more  complex 

*  It  might  be  asserted  with  some  truth  tiiat  the  first  procedure 
of  the  mind  is  synthetic :  there  must  be  a  "whole" — a  synthesis — 
however  vague,  for  analysis  to  work  upon.  Ctrtainly  the  la&t  pro- 
cedure of  the  mind  is  "  synthetic." 


•i 


'  I    I 


I   u 


■W 


!      ,  .  1.1 


102 


THE  PSYCHOLOGY  OF  NUMBER. 


>'.  * 


i!i 


i 


than  subtraction,  the  inverse  of  addition.  We  may,  as 
we  liave  seen,  add  a  number  of  threes,  for  example — 
giving  eacli  addend  a  momentary  attention,  and  then 
dropping  it  utterly  from  consciousness — without  grasp- 
ing the  /actor,  which,  with  three  as  the  other  factor, 
will  give  a  p?'odu€t  equal  to  the  smn  of  the  addends. 
So  in  division,  the  inverse  operation,  this  factor  does 
not  come  merely  from  the  successive  subtractions  of 
three  from  the  sum  until  there  is  no  remainder ;  here, 
as  in  addition,  a  further  mental  operation  is  necessary 
before  the  factors  are  discovered — that  of  counting  the 
times  of  repetition  ;  i.  e.,  of  finding  the  ratio  of  the  sum 
(dividend)  to  the  repeated  subtrahend. 

We  are  told,  too,  that  when  we  separate  8  cubes  into 
4  equal  parts  it  is  instantly  seen  that  8  contains  2  four 
times,  that  2  is  one  fourth  of  8,  that  2  may  be  taken 
four  times  from  8,  and  that  these  results  being  obtained 
independently  of  addition  and  multiplication,  division 
and  subtraction  may  be  taught  first. 

There  seems  to  be  a  fallacy  lurking  liere.  "VYe  may, 
indeed,  separate  8  cubes  into  two  parts,  or  four  parts, 
or  eight  parts ;  but  that  is  mere  physical  separation. 
Granting  recognition  of  the  concrete  (spatial)  element 
• — the  measuring  units — how  does  the  abstract  element 
— the  idea  of  times — arise?  How  do  we  know  that 
there  is  four  times  2  or  eight  times  1  ?  Only  by  count- 
ing, by  relating,  by  an  act  of  synthesis — the  last  pro- 
cedure of  the  mind  in  a  complete  process  of  thought. 
Thus,  the  fallacy  referred  to  ignores  one  of  the  two 
necessary  factors  (relation)  in  the  psychical  process  of 
number.  It  must  presuppose  that  counting  does  not 
imply  addition  and  multiplication.     What  is  counting 


/ 


THE  ARITHMETICAL  OPERATIONS. 


103 


/ 


but  addition  by  ones  ?  What  is  five,  if  not  one  more 
than  four ;  and  four,  if  not  one  less  than  five  ?  How  is 
four,  e.  g.,  defined  except  as  that  number  which,  ap- 
pUed  to  a  unit  of  measure,  denotes  a  quantity  consist- 
ing of  three  such  units  and  one  unit  more  ?  This  count- 
ing, which  begins  with  discrete  quantity  (collection  of 
objects)  in  the  first  stage  of  measurement,  is  addition 
(with  subtraction  implied)  by  ones,  and  the  idea  of  mul- 
tiplication and  division  involved  in  it  becomes  evolved 
(in  counting  with  an  exact  unit  of  measure)  with  the 
growth  of  numerical  abstraction  and  the  consequent 
development  of  the  measuring  power  of  number. 

It  seems  plain,  then,  that  in  the  development  of  num- 
ber as  the  instrument  of  measurement  there  is  first  the 
rational  use,  leading  to  conscious  recognition,  of  the 
aggregation  idea — that  is,  addition  and  subtraction ; 
then  the  definite  use,  leading  to  conscious  recognition, 
of  the  factor  (times)  idea — that  is,  multiplication  and 
division.  In  other  words,  the  psychological  order  as 
determined  by  the  demand  on  conscious  attention  is 
the  old-time  arrangement — Addition  and  Subtraction, 
Multiplication  and  Division. 

It  is  the  order  in  which  numerical  ideas  and  pro- 
cesses appear  in  the  evolution  of  number  as  the  instru- 
ment of  measurement ;  the  order  in  which  they  appear 
in  the  reflective  consciousness  of  the  child ;  the  order 
of  increasing  growth  in  psychological  complexity.  This 
order  may  be  said  to  reverse  the  order  of  logical  de- 
pendence, but  the  psychological  order  rather  than  that 
of  logical  dependence  is  to  be  the  guide  in  teaching. 

N'ot  Exclusive  Attention  to  One  Hide  or  Process. — 
But  the  true  method,  as  based  on  this  psychological 


m 


"I  I. 


s 


104 


THE  PSYCHOLOGY  OF  NUMBER. 


',  i 


I;    I 


m .  ■' 


■n 


order  of  instruction,  by  no  means  implies  that  addition 
and  subtraction  are  to  be  completely  mastered  before 
the  introduction  of  any  multiplication  or  division  or 
fractions.  Quite  the  contrary.  On  account  of  their 
greater  complexity  the  higher  processes  are  not  to  be 
taught  analytically — made,  that  is,  an  object  of  con- 
scious attention  from  the  first ;  but  they  may  and  should 
be  freely  used,  and  thus  relieve  the  monotony  of  too 
much  addition  and  subtraction,  and  at  the  same  time 
prepare  the  way  for  their  conscious  (analytic)  use. 

Because  of  the  rhythmic  character  of  multiplica- 
tion such  forms  of  it  as  can  be  objectively  presented 
in  simple  constructions — the  putting  together  of  tri- 
angles, squares,  cubes,  etc.,  to  make  larger  or  more 
complex  figures,  of  dimes  to  make  dollars — are  much 
more  easily  learned  than  many  of  the  addition  and  sub- 
traction combinations.  The  ideas  of  ratio  should  be  in- 
cidentally introduced  in  connection  with  certain  values 
(e.  g.,  9,  12,  6,  16,  100,  etc.)  practically  from  the  begin- 
ning ;  and  consequently  the  process  of  fractions  in  sim- 
ple forms,  and  its  symbolic  statement.  Kothing  but  the 
demands  of  a  preconceived  theory  could  so  nullify  ordi- 
nary common  sense  as  to  suppose  that  there  is  no  alter- 
native between  either  exhausting  all  operations  with 
every  number  before  going  on  to  the  next  higher,  or 
else  mastering  all  additions  and  sul)tractions  before  go- 
ing on  to  ratio — multiplication  and  division.  Practical 
common  sense  and  sound  psychology  agree  in  recom- 
mending first  the  e?nphasis  on  addition  and  subtraction, 
with  incidental  introduction  of  the  more  rhythmical  and 
obvious  forms  of  ratio,  and  gradual  change  of  emphasis 
to  the  processes  of  multiplication  and  division.     If  the 


\ 


THE  ARITHMETICAL  OPERATIONS. 


105 


idea  of  number  as  a  mode  of  measurement  is  followed, 
it  will  be  practically  impossible  to  w^ork  in  any  other 
way.  Even  while  working  explicitly  w^th  addition  and 
subtraction — inches,  feet,  ounces,  pounds,  dollars,  cents, 
etc. — the  process  of  ratio  is  constantly  being  introduced. 
The  child  can  not  help  feeling  that  1  inch  is  one  third 
of  8  inches,  10  cents  (1  dime)  one  tenth  of  a  dollar,  etc. ; 
and  this  natural  growth  towards  the  definite  conception 
of  ratio  is  only  checked,  not  forwarded,  by  compelling 
a  premature  conscious  recognition  of  the  nature  of  the 
process. 

Addition  and  Subtraction. — The  general  nature  of 
these  operations  as  concerned  with  measurenient  through 
the  process  of  aggregating  minor  units  or  parts  has  al- 
ready been  dealt  with.  Two  or  three  points  may,  how- 
ever, be  considered  in  more  detail. 

1.  Work  from  and  within  a  Whole. — Here,  as  every- 
where, the  idea  of  a  magnitude — a  whole  of  quantity — 
corresponding  to  some  one  unified  activity  should  be 
present  from  the  first.  Some  vague  quantity  or  whole, 
which  is  to  be  measured  by  the  putting  together  of  a 
number  of  parts,  alone  gives  any  reason  for  performing 
the  operation  and  sets  any  limit  to  it.  The  process  of 
breaking  up  the  whole  into  parts  and  then  putting  to- 
gether these  parts  into  a  whole,  measures  or  defines 
what  was  originally  a  vague  magnitude  and  gives  it 
precise  numerical  value.  In  dealing,  say,  with  6,  we 
may  begin  with  a  figure  like  this  ^."^     This  is  a  unity 

or  whole — it  is  one.     But  its  value  is  indefinite.     The 


*  This  may,  of  course,  be  constructed  out  of  splints,  or  whatever 
is  convenient. 


ii 


106 


THE  PSYCHOLOGY  OF  NUMBER. 


Ill 


k: 


counting  off  of  the  various  sticks  changes  the  vague  unity 
into  a  measured  unity,  but  these  parts  always  fall 
within  the  orujinal  unity.  Thei*e  is  always  a  sense  of 
the  whole  connecting  them  together.  If  the  square  lias 
already  been  mastered,  the  figure  will  be  recognised 
as  one  4  -\-  one  2.  Or,  if  one  of  the  diagonals  is  changed 
thus  \/y^,  it  will  be  recognised  as  two  triangles — that  is, 
as  one  3  -|-  one  3.  Or,  of  course,  it  may  be  taken  all 
to  pieces  and  put  together  again  and  recognised  as  6 
parts  of  the  value  of  1  each.  Or,  the  pupil  may  be 
told  to  make  "pickets"  or  "tents"  of  the  figure,  and, 
arranging  them  as  follows,  A  A  A  ,  see  that  there  are 
three  groups  of  the  value  of  2  each. 

The  principle  kept  in  mind  in  this  instance  is  that  of 
the  equation  and  its  rhythmic  construction,  {a)  Accord- 
ing to  the  prevalent  method,  six,  when  reached,  would 
be  simply  six  ones,  six  separate  unities,  that  is — not,  as 
in  the  foregoing  illustration,  six  parts  of  unit  value  each. 
No  matter  how  much  the  teacher  is  urged  to  have  the 
pupils  recognise  six  at  a  glance,  and  not  count  up  the 
various  unities  in  it  separately,  still  the  fact  remains 
that  it  can  not,  by  that  method,  be  grasped  as  a  whole ; 
while  by  the  psychological  method  it  can  not  be  grasped 
in  any  other  way.  (?>)  It  is  also,  upon  the  psychologic- 
al method,  regarded  as  having  a  value  equal  to  (meas- 
ured by)  its  constituent  minor  wholes.  We  are  alwavs 
^>i  .-■,,<; -;^-  vjiihin  a  value,  simply  making  it  more  clear 
.ij  1  detinite,  not  blindly  or  vaguely  from  fixed  unities 
to  their  accidental  sum — accidental,  that  is,  so  far  as 
tiie  action  of  mind  is  concerned.  As  a  result,  the  psy- 
chological method  appeals  directly  to  the  power  of  break- 
ing up  a  larger  whole  into  minor  wholes,  and  putting 


THE  ARITHMETICAL  OPERATIONS. 


107 


these  together  to  make  a  larger  whole.  It  appeals  to  the 
constructive  rhythmic  interest,  never  to  mere  memoriz- 
ing. It  gives  the  maximum  opportunity  for  the  exer- 
cise of  power ;  it  leaves  the  minimum  for  mere  mechan- 
ical drill.  Because,  dealing  with  wholes,  intuition  may 
be  used  ;  the  rationality  of  the  principle — the  construe- 
tlon  of  a  complete  whole  hy  means  of  jpartial  wholes — 
may  be  objectively  seen  and  clearly  appreciated. 

It  may  be  laid  down,  then,  in  the  most  emphatic 
terms,  tliat  the  value  of  any  device  for  teaching  addi- 
tion depends  upon  whether  or  not  it  begins  with  a  whole 
which  may  he  intuitively  presented^  and  whether  or  not 
it  proceeds  by  the  rhythmic  partition  of  this  original 
whole  into  minor  wholes,  and  their  recombination. 

2.  Use  of  Subtraction  as  Inverse  Operation. — Upon 
this  basis  the  process  of  subtraction  is  always  iised  simulta- 
neously with  addition.  In  beginning  with  a  fixed  unity, 
or  an  aggregate  of  such  unities,  the  ''  method  "  may  tell 
us  to  teach  addition  and  subtraction  together  (or,  what 
is  really  meant,  one  immediately  after  the  other),  but 
they  can  not  be  employed  at  the  same  time.  If  1  is  one 
thing,  2  two  things,  3  three  things,  and  so  on,  it  requires 
one  mental  act  to  unite  two  or  more  such  things,  and 
notice  the  resulting  sum  ;  and  another  act  to  remove 
one  or  more,  and  note  the  resulting  diiierence.  But  in 
beginning  with  ^  and  noting  that  it  is  made  up  of  Q, 
or  4,  and  X,  or  2,  the  synthesis  (recognition  of  the  whole 
of  parts)  and  analysis  (recognising  the  parts  in  the  w^iole) 
are  absolutely  simultaneous.  It  is  one  and  the  same  act 
(6  =  4-f-  2),  which  becomes  in  outward  statement  addi- 
tion or  subtraction,  according  as  the  emphasis  is  directed 
upon  both  of  the  parts  equally,  or  upon  the  whole  and 


■    1 


(' 


I  1. 


■if 


m 


'"i 


u 


«:." 


S'    I 


i\     -ii 


108 


THE  PSYCHOLOGY  OF  NUMBER. 


'!>' 


It       ■ 

|1    " 


Mi 


^iii 


;.-.  t 


one  of  the  parts.  If,  for  example,  in  the  above  in- 
stance the  Q  and  the  X  are  both  equally  familiar, 
then  the  construction  would  probably  appeal  to  the 
child  as  addition,  putting  together  the  more  familiar  to 
make  the  more  unfamiliar.  But  if  the  []]  alone  is  very 
familiar,  he  might  rather  notice  that  the  dijfcrenee 
between  the  square  and  the  original  whole,  namely, 
X,  or  2  units. 

3.  The  Conscious  Process  of  Subtraction  slightly 
more  Complex. — The  conscious  recognition  of  subtrac- 
tion, however,  is  a  slightly  more  complex  process — 
makes  more  demand  upon  attention — than  the  con- 
scious interpretation  of  addition.  In  addition,  the 
whole  emphasis  is  upon  the  result;  it  is  not  necessary 
to  keep  the  parts  separate  at  all.  The  sum  of  5  and  4, 
e.  g.,  is  first  of  all  supplied  by  intuition,  and  where 
the  association  is  complete  the  mind  merely  touches,  as 
it  were,  the  symbols,  and  the  sum  appears  in  conscious- 
ness. If,  for  example,  we  know  that  James  and  John 
and  Peter  have  a  certain  amount  of  money — the  unde- 
fined whole — of  which  James  has  6  and  John  8  and 
Peter  12  cents,  we  instantly  merge  or  absorb  each  pre- 
ceding quantity  in  the  next  greater — 6, 14,  26.  As  soon 
as  the  two  parts  are  added  they  are  dropped  as  separate 
parts,  the  resulting  whole  is  alone  kept  in  mind.  But 
in  subtraction  it  is  necessary  to  note  both  the  whole  and 
the  given  part,  and  the  relation  between  them.  If  we 
say  that  of  the  total  amount  *  James  has  6  cents,  John 

*  While  it  is  not  necessary  always  to  introduce  the  idea  of  the 
total  first  in  words,  it  should  be  done  even  verballv  until  we  are 
sure  that  the  child's  mind  always  supplies  the  idea  of  a  whole 
from  and  within  which  he  is  constantly  working. 


W  i  = 


THE   ARITHMETICAL  OPERATIONS. 


109 


I  -1 


2  more  than  James,  and  Peter  4  more  than  John,  then 
the  addition  problem  re(|uires  the  same  attention  to  the 
two  terms  separately  and  to  the  result  as  is  required  in 
subtraction.  There  is  the  idea  of  definiteness  or  relative 
moreness,  and  not  merely  the  idea  of  an  aggregate  more- 
ness.  Here,  as  in  subtraction,  we  are  approaching 
nearer  to  ratio. 

Multiplication  :  Genesis  of  The  Factor  Idea. 

We  have  seen  that,  though  multiplication  is  not  iden- 
tical with  addition  (even  with  the  special  case  of  addi- 
tion where  the  addends  are  all  equal),  it  has  its  genesis 
in  addition,  taking  its  rise  in  counting^  which  is  the 
fundamental  numerical  operation.  Counting  is  the  re- 
lating process  in  the  mental  activity  which  transforms 
an  indefinite  whole  of  quantity  into  a  definite  whole. 
It  begins  with  discrete  quantity,  and  is  first  of  all  largely 
mechanical — an  operation  with  things.  The  child  in  his 
first  countings  does  not  consciously  relate  the  things ; 
his  act  is  not  one  of  rational  counting.  He  is  apt  to 
think  that  the  number-names  are  the  names  of  things ; 
that  three^  e.  g.,  is  not  the  third  of  three  related  things, 
l)ut  the  nam^e  of  the  third  thing ;  and  on  being  asked 
to  take  up  three  he  will  fix  upon  the  single  thing  which 
in  counting  was  called  three. 

But  starting  with  groups  of  objects  and  repeating 
the  operations  of  parting  and  wholing,  he  soon  begins 
to  feel  that  the  objects  are  related  to  one  another  and 
to  the  whole.  This  is  a  growth  towards  the  true  idea 
of  number,  but  the  idea  is  not  yet  developed.  There  is 
a  relating,  but  not  the  relating  which  constitutes  num- 
ber.    In  the  process  of  counting  one,  two,  etc.,  getting 


»i 


\  w 


I  i 


i  I 
1 1 


!      f1 


i'-  -i^ 


■  ^  ■■If 


I 


no 


THE  rSYCriOLOGY  OF  NUMBER. 


4 

t, 

t 

[ 

ml' 

i 

ijt  ; 


11 

fel 


lu 


as  far  as  five,  e.  g.,  lie  is  conscious  that  five  is  connected 
with  what  goes  l)efore.  This  perception  is  one  of  inore- 
ness  or  lessness,  of  aggregation  ;  live  is  more  than  four, 
and  at  last,  detinitelj,  it  is  one  more  than  four.  With 
the  continuance  of  tlie  physical  acts  there  is  further 
growth  towards  the  higher  conception.  He  separates 
a  whole  into  parts  and  remakes  the  whole :  he  combines 
(using  intuitions)  unequal  groups  of  measuring  units 
(e.g.,  3  feet  and  4  feet)  to  express  them  as  ories ^'  he 
counts  by  o?ies,  (/roups  of  two  things,  of  three  things, 
etc.,  and  at  last  the  idea  of  times,  of  pure  number,  is 
definitely  grasped.  The  "five"  is  no  longer  mereli/ 
one  more  tlian  four,  it  is  five  times  one,  whatever  that 
one  may  be.  In  other  words,  he  has  passed  from  the 
lower  idea  to  the  higher ;  from  the  idea  of  mere  aggre- 
gation to  that  of  times  of  repetition ;  from  addition  to 
multiplication. 

It  is  plain  that  there  must  be  time  for  the  develop- 
ment of  this  abstracting  and  generalizing  power.  In 
fact,  the  complete  development  of  the  "  times  "  idea, 
this  factor  relation,  corresponds  with  the  stages  of  the 
measuring  power  of  number.  The  higher  power  of 
numerical  abstraction  is  the  higher  power  of  the  tool 
of  measurement.  This  normal  growth  in  the  power  of 
abstracting  and  relating  can  not  be  forced  by  any — the 
most  minute  and  ingenious — analyses  on  the  part  of  the 
teacher.  The  learner  may  indeed  be  drilled  in  such 
analyses,  and  may  glibly  repeat  as  well  as  "  reason  out " 
the  processes ;  just  as  he  can  be  drilled  to  the  repeti- 
tion of  the  words  of  an  unknown  tongue,  or  any  other 
product  of  mere  sensuous  association.  But  it  does  not 
follow  that  he  knows  number,  that  be  has  grasped  the 


I 


''i  \ 


•  i 


fl 


1 


THE  ARITHMETICAL  OPERATIONS. 


Ill 


idea  of  times.  The  difficulty  is  not  in  the  word 
times,  as  some  appear  to  think  ;  it  is  in  the  idea 
itself,  and  would  not  disappear  even  if  the  word  were 
(as  some  propose)  exorcised  from  our  arithmetics. 
It  has  not  yet  been  proposed  to  eliminate  the  idea 
itself — i.  e.,  the  idea  of  number — from  the  science  of 
number. 

Summary. — (1)  Counting  is  fundamental  in  the  de- 
velopment of  numerical  ideas ;  as  an  act  or  operation 
with  objects  it  is  at  first  largely  a  mechanical  process, 
but  with  the  increase  of  the  child's  power  of  abstrac- 
tion it  gradually  becomes  a  rational  process.  (2)  From 
this  (partly)  physical  or  mechanical  stage  there  is  evolved 
the  relation  of  more  or  less,  and  addition  and  subtrac- 
tion arise — that  is,  e.  g.,  five  is  one  more  than  four. 
(3)  The  addition,  through  intuitions,  of  unequal  (meas- 
ured) quantities,  which  are  thus  conceived  and  expressed 
as  a  defined  unity  of  so  many  ones,  is  an  aid  to  the  de- 
velopment of  the  times  idea.  (4)  Continuance  of  such 
operations — appealing  to  both  eye  and  ear — brings  out 
this  idea  more  definitely — e.  g.,  five  is  not  now  simply 
one  more  than  four,  it  is  five  times  one.  (5)  Counting 
(by  ones)  groups  of  twos,  threes,  etc.,  brings  out  still 
more  clearly  the  idea  of  times.  (6)  Through  repeated 
intuitions,  sums  (the  results  of  uniting  equal  addends) 
become  associated  with  times,  the  factor  idea  (times  of 
repetition)  displaces  the  part  idea  (aggregation),  and 
multiplication  as  distinct  from  addition  arises  explicitly 
in  consciousness. 

The  Process  of  Multiplication. — The  expression  of 
measured  quantity  has,  we  have  seen,  two  components, 
one  denoting  the  unit  of  measure,  and  the  other  de- 


I  h 


1 1 


\-  h 


v'--\ 

^  h-l 

^  ^% 

urn 

I  ill'.-;;,*! 


w 


112 


TIIK  PSYCIIOLCKJY   OF   NUMBER. 


if' 


;M 


'li'i 


noting  the  number  of  tliese  units  constituting  the  ({uan- 
tity.  But  since  the  unit  of  nieasure  is  itself  composed 
of  a  definite  num])er  of  parts — is  definitely  measured 
by  some  other  unit — it  is  clear  that  we  actually  con- 
ceive of  the  quantity  as  made  up  of  so  many  given  units 
(direct  units  of  measure),  each  measured  by  so  many 
minor  units.  For  convenience  we  may  call  these  minor 
units  "  primary,"  as  making  up  the  direct  unit  of  meas- 
ure, and  this  direct  unit,  as  being  made  up  of  primary 
units,  may  be  called  the  ''  derived  "  unit.  We  shall  thus 
have  in  the  complete  expression  of  any  measured  quan- 
tity, (1)  the  derived  unit  of  measure,  (2)  the  number  of 
such  units,  and  (3)  the  number  of  primary  units  in  the 
derived  unit  of  measure. 

For  example,  take  the  following  expressions  of  quan- 
tity :  In  a  certain  sum  of  money  there  are  seven  counts 
of  five  dollars  each ;  here  the  derived  unit  of  measure 
is  Jive  dollars,  the  number  of  them  is  seve?i,  and  the  pri- 
mary unit  is  one  dollar.  The  cost  of  a  farm  of  sixty 
acres  at  fifty  dollars  an  acre  is  sixty  fifties ;  hei'e  the 
derived  unit  of  measure  is  fifty  dollars,  the  number  of 
them  sixty,  and  the  primary  unit  one  dollar.  The  length 
of  a  field  is  fifteen  chains — that  is  (in  yards),  fifteen 
twenty-twos ;  here  the  derived  unit  is  twenty-two  yards, 
the  number  of  them  fifteen,  and  the  primary  unit  one 
yard.  In  the  quantity  $fxf  the  primary  unit  is  $1, 
the  derived  unit  ${,  and  f  is  the  nwnher  expressing  the 
quantity  in  terms  of  the  derived  unit. 

Now,  when  a  quantity  is  expressed  in  terms  of  the 
derived  unit,  it  is  often  necessary  or  convenient  to  ex- 
press it  in  terms  of  a  primary  unit.  Thus,  in  the  fore- 
going examples,  the  sum  of  money  expressed  as  seven 


■ 


\ 


m 


TIIK   AIUTIIMETICAL  OPERATIONS. 


113 


fives  may  be  expressed  as  thirty-fve  ones;  the  cost  of 
tlie  farm,  expressed  as  sixty  fifties,  may  he  expressed  as 
three  thousand  ones ;  and  tlie  length  of  the  field,  ex- 
pressed as  jifteoi  twenty-twos,  may  he  expressed  as 
three  hundred  and  thirty  ones  (yards);  and  ^fxf  is 
measured  hy  ig^^  in  terms  of  the  primary  unit,  in  each 
of  these  cases  the  second  expression  of  the  measured 
quantity  merely  states  explicitly  the  number  of  minor 
(or  primary)  units  which  is  implied  in  the  first  ex- 
])ression.  The  operation  by  which  we  find  the  number 
of  primary  units  in  a  quantity  expressed  by  a  given 
number  of  derived  units  is  Multiplication.  It  is  plain 
that  the  idea  of  times  (pure  number,  ratio)  is  prominent 
in  this  operation ;  we  have  the  times  the  i^rimary  unit 
is  taken  to  make  up  the  derived  unit,  and  the  times  the 
derived  unit  is  taken  to  make  up  the  quantity.  The 
multiplicand  always  rei)resents  a  number  of  (primary) 
units  of  quantity ;  the  multiplier  is  always  pure  num- 
ber, representing  simply  the  times  of  repetition  of  the 
derived  unit.  But  from  the  nature  of  the  measuring 
process  the  two  factors  of  the  product  may  be  inter- 
changed, the  times  of  repetition  of  the  primary  unit 
may  be  commuted  with  the  times  of  repetition  of  the 
derived  unit;  in  other  words,  the  7zw??^j6^/' which  is  ap- 
plied to  the  primary  unit  may  be  commuted  with  the 
numher  whicli  is  applied  to  the  derived  unit. 

Correlation  of  Factors. — In  our  conception  of  meas- 
ured quantity  these  two  ideas  are,  as  has  been  shown, 
absolutely  correlative.  Measuring  a  line  of  tioelve  units 
by  a  line  of  tvm  units,  the  numerical  value  is  six  /  if  we 
consciously  attend  to  the  process,  the  related  conception 
instantly  arises ;  we  can  not  think  six  times  two  units 


[!! 


"■I 


'  ij 


\{ 


[i 


114 


THE  PSyCHOLOGY  OP  NUMBER. 


'li 


jl 


•^1 


;lt 


(i 


)  '■ 


il 


without  tliinking  two  times  six  units,  because  we  can  not 
think  one  unit  six  times  without  thinking  one  whole  of 
six  units.  So,  in  measuring  a  rectangle  8  inches  long 
by  10  inches  wide,  we  can  not  analytically  attend  to  the 
process  which  gives  the  result  of  8  square  inches  taken 
ten  times  w^ithout  being  conscious  of  the  inevitable 
correlate,  10  square  inches  taken  eight  times.  In  gen- 
eral :  To  think  the  measurement  of  any  quantity  as  h 
units  taken  a  times,  is  to  think  its  correlate  a  units 
taken  h  times;  for  h  units  is  h  times  one  unit,  and  every 
one  in  h  is  repeated  a  times,  giving  a  units  once,  a  units 
twice,  etc. — that  is,  a  units  b  times. 

Educational  Applications. 

1.  Just  as,  in  addition,  we  must  always  begin  with  a 
vague  sense  of  some  aggregate,  and  then  go  on  to  make 
that  definite  by  putting  together  the  constituent  units, 
while  in  subtraction  we  begin  with  a  defined  aggregate 
and  a  given  part  of  it,  and  go  on  to  determine  the  other 
parts  ;  so,  in  multiplication,  we  begin  with  a  compara- 
tively vague  sense  of  some  whole  which  is  to  be  more 
exactly  determined  by  the  "  product,"  w^hile  in  division 
we  begin  w^th  an  exactly  measured  whole,  and  go  on  to 
determine  exactly  its  measuring  parts.  In  multiplica- 
tion the  order  is  as  follows  :  (1)  The  vague  or  imper- 
fectly defined  magnitude  ;  (2)  the  definite  unit  of  value 
(primary  unit),  which  has  to  be  repeated  to  make  the 
derived  (direct)  unit  of  measure — the  multiplicand  ;  (3) 
the  number  of  times  this  derived  unit  is  to  be  repeated 
— the  multiplier  ;  and  (4)  the  product — the  vague  mag- 
nitude now  definitely  measured. 

The  operation  of  multiplication,  therefore,  already 


THE  ARITHMETICAL  OPERATIONS. 


115 


t 


s 


implies  division  /  the  definite  unit  of  measurement 
which  constitutes  the  multiplicand  is  plvvajs  a  certain 
exact  (equal)  portion  of  some  whole.  Hence  multipli- 
cation always  implies  ratio  ;  the  whole  magnitude  bears 
to  the  unit  of  measure  a  ratio  which  is  expressed  in  the 
number  of  times  (represented  by  the  multiplier)  the 
unit  has  to  be  taken  to  measure  that  magnitude — to 
give  it  accurate  numerical  value.  In  fact,  the  process ' 
is  simply  one  of  changing  the  numher  which  measures 
a  magnitude  by  changing  the  unit  of  measure — i.  e.,  by 
substituting  for  the  given  unit  of  measure  the  primary 
unit  from  which  it  was  derived.* 

2.  In  multiplication,  then,  as  in  addition,  we  are  not 
performing  a  purposeless  operation,  or  one  with  unre- 
lated parts  and  isolated  units  ;  rather,  we  begin  and  end 
with  some  magnitude  requiring  measurement,  keeping 
in  mind  that  what  distinguishes  multiplication  is  the 
kind  of  measurement  it  uses — that,  namely,  in  which  a 
unit  itself  measured  off  by  other  units  is  taken  a  certain 
number  of  times. 

3.  The  psychology  of  number,  therefore,  impera- 
tively demands  that  the  quantity  which  is  to  be  finally 
expressed  by  the  "  product "  should  first  be  suggested,, 
just  as  in  addition  the  quantity  given  by  "  sum,"  within 
which  and  towards  which  we  are  working,  is  kept  in 


:   t     J 
1     [jS 

r 


i   Hi| 
i   I 


;jr 


*  In  such  instances  as  multiply  7  apples  by  4,  the  idea  of  exact 
division  or  ratio  is  not  so  evident,  but  the  7  apples  must  be  taken 
as  one  of  four  equal  portions — i.  e.,  as  having  the  ratio  i  to  the 
whole  quantity.  The  fact,  however,  that  the  idea  of  an  exact  unit  of 
measurement  is  not  so  clearly  present,  is  a  strong  reason  for  using 
fewer  examples  of  this  sort,  and  more  of  those  involving  standard 
units  of  measure. 


vu 


116 


THE  PSYCHOLOGY   OP  NUMBER. 


P'l 


ni 


.', 


mind  from  the  first.  If  the  child  sees,  e.  g.,  tliat  there 
is  a  certain  field  of  given  dimensions  whose  area  is  to 
be  ascertained,  or  a  piece  of  cloth  of  given  lengtli  and 
price  per  yard,  of  which  the  cost  is  to  be  determined, 
the  mind  has  something  to  rest  upon,  a  clearly  defined 
purpose  to  accomplish.  Beginning  with  a  more  or  less 
definite  image  of  the  thing  to  be  reached,  the  subsequent 
steps  have  a  meaning,  and  the  entire  process  is  rational 
and  consequently  interesting.  But  when  he  is  asked  how 
much  is  4  times  8  feet,  or  9  times  32  cents,  there  is  no 
intrinsic  reason  for  performing  the  operation  ;  psycho- 
logically it  is  senseless,  because  there  is  no  motive,  no 
demand  for  its  performance.  The  sole  interest  which 
attaches  to  it  is  external,  as  arising  from  the  mere  ma- 
nipulation of  figures.  Under  an  interested  teacher,  in- 
deed, even  the  pure"  figuring"  work  may  be  interest- 
ing ;  but  this  interest  is  re-enforced,  transfoi'med,  when 
the  mechanical  work  is  felt  to  be  the  means  by  which 
the  mind  spontaneously  moves  by  definite  steps  towards 
a  definite  end.  This  does  not  mean,  we  may  once  more 
remark,  that  examples  like  8  feet  x  4,  or  even  8x4, 
are  to  be  excluded,  but  only  that  the  haJnt  of  regard- 
ing number  as  measuring  quantity  should  be  perma- 
nently formed.  The  ])upil  should  be  so  trained  that 
all  addends,  sums,  minuends,  products,  multiplicands, 
dividends,  quotients,  could  be  instantly  interpreted  in 
tlieir  nature  and  function  as  connected  with  the  process 
of  measurement.  For  example  :  A  farmer  has  8  bush- 
els of  potatoes  to  sell,  and  the  market  price  is  55  cents 
a  bushel :  how  much  can  he  get  for  them  ? 

This  and  similar  examples  are  often  presented  in 
such  a  way  that  when  the  pupil  gets  the  product,  $4.40, 


THE  ARITHMETICAL   OPERATIONS. 


117 


r^ 

t 


his  mind  stops  short  with  the  mere  idea  of  the  product 
as  a  series  of  figures.  This  is  irrational ;  $4.40  in  itself 
is  not  a  product ;  no  quantity  or  value  is  ever  in  itself 
a  product ;  but  as  a  product  it  measures  more  definitely 
the  value  of  some  quantity.  In  other  words,  the  prod- 
uct must  always  be  interpreted  /  it  must  be  recognised 
as  the  accomplished  measurement  of  a  measured  quan- 
tity in  terms  of  more  familiar  or  convenient  units  of 
measure. 

4.  The  multiplicand  must  always  be  seen  to  be  a 
unit  in  itself,  no  matter  how"  lai-ge  it  is  as  expressed 
in  minor  units.  It  signifies  the  known  value  of  the 
unit  with  which  one  sets  out  to  measure  ;  it  is  the  meas- 
uring rod,  as  it  were,  which  is  none  the  less  (rather  the 
more)  a  unit  because  it  is  defined  by  a  scale  of  parts.  A 
foot  is  none  the  less  one  because  it  may  be  written  as 
12  inches  or  as  192  sixteenths  ;  nor  is  a  mile  any  the  less 
a  unit  because  it  is  written  as  320  rods  or  as  5280  feet. 
The  ineradicable  defect  of  the  Grube  method,  or  any 
method  which  conceives  of  a  unit  as  one  thing  instead 
of  as  a  standard  of  measuring,  is  that  it  can  never  give 
the  idea  of  a  multiplicand  as  just  one  unit — a  part  used 
to  measure  a  whole. 

5.  It  is  important  so  to  teach  from  the  beginning  that 
a  clear  and  definite  conception  of  the  relation  between 
parts  and  times  may  be  developed.  Of  course,  nothing 
is  said  till  the  time  is  ripe  about  tlie  law  of  "commu- 
tation "  ;  but  the  idea  should  be  present,  and  should 
be  freely  used.  If  a  quantity  of  12  units  is  meas- 
ured by  3  units  repeated  four  times,  the  child  can  be 
led  to  see — will  probably  discover  for  himself — that 
this  measurement  is   identical  with  the  measurement 


1     i 


J    :!:  -i 


Il '  > 


Vv 


m 


s  " 


!i'i 


Mii 


i  • 


t'<i  i 


118 


THE  PSYCHOLOGY  OF  NUMBER. 


4  units  repeated  three  times.  Katioiially  using  this  idea 
of  commutation  in  repeated  operations,  the  child  will 
soon  get  possession  of  a  principle  by  which  he  can 
easily  interpret  both  processes  and  results  in  numerical 
work. 


I 


. ':  I  f 


'  <j '-I 


t 
•t 


CHAPTEE  YII. 

numerical  operations  as  external  and  as  intrinsic 

to  number. 

Division  and  Fractions. 

Division. — As  multiplication  has  its  genesis  in  addi- 
tion, but  is  not  identical  with  it,  so  division  has  its  gene- 
sis in  subtraction,  but  is  not  identical  with  it.  Just  as 
multiplication  comes  from  the  explicit  association  of 
the  numher  of  equal  addends  with  their  sum,  and  the 
substitution  of  the  factor  idea  (ratio)  for  the  part  idea, 
so  division  comes,  in  the  last  analysis,  from  the  explicit 
association  of  the  number  of  equal  subtrahends  from 
the  same  sum  (dividend),  and  the  substitution  of  the 
factor  idea  for  the  part  idea.  In  other  words,  division 
is  the  inverse  of  multiplication,  just  as  subtraction  is 
the  inverse  of  addition.  Further,  as  in  multiplication, 
both  factors  are  the  expression  of  a  measured  quantity 
and  are  interf'hangeable,  so  in  division  either  of  the  fac- 
tors (divisor  and  quotient)  which  produced  the  dividend 
can  be  commuted  with  the  other.  In  multiplication,  for 
example,  we  have  4  feet  x  5  =  5  feet  x  4  =  20  feet ; 
and  the  inverse  problem  in  division  is,  given  the  20 
feet,  and  either  of  the  factors,  to  find  the  other  factor. 
We  solve  the  problem  not  by  subtraction,  but  by  the 
use  of  the  factor,  or  ratio,  idea. 

119 


i  i' 


11-^ 


!.     ,  '. 


mw: 


1! 


120 


THE    PSYCHOLOGY   OF   NUMBER. 


>  > 


Ji 


In  multiplication,  as  already  snggested,  we  may  look 
at  a  product  of  two  factors  in  two  ways :  For  example, 
20  feet  =  4  feet  x  5  =  also  5  feet  x  4,  or  five  times 
four  times  1  foot  =  four  times  five  times  1  foot — that 
is,  we  may  nse  the  primary  unit  of  measure  "  1  foot " 
with  either  tlip  f;  .  •  times  or  the  five  times.  Or,  stated 
in  general  terms,  1/  times  a  times  the  primary  unit  of 
measure  is  identical  with  a  times  h  times  this  primary 
unit — that  is,  we  may  interchange  at  pleasure  the  nu- 
merical value  of  '^'.c  rr,p,o,suring  imit  (the  derived  unit 
as  made  up  of  pr.ir^;.  ;-  uTiits)  with  the  numerical 
value  of  the  whole  quaiuity  as  made  up  of  these  de- 
rived units.     This  is  in^nort'   "  as  interpreting  the  pro- 


cess and  result  in  division. 


J  r 


'ave  20  feet  and  the 


factor  4  feet  given  to  find  the  other  factor,  we  use  the 
measurement  4  feet  x  5  =  20  feet.  If,  on  the  other 
hand,  we  have  the  20  feet  and  the  numljer  5  given  to 
find  the  other  factor,  we  may  use  eitJier  measurement; 
we  may  divide  directly  by  the  number  5,  or  we  may 
"  concrete  "  the  5  (consider  it  as  denoting  5  feet),  and 
get  the  other  factor  4  (times) ;  for  we  know  that  4 
times  5  feet  is  identical  with  5  times  4  feet,  and  the 
conditions  of  the  question  require  tlie  latter  interpre- 
tation. In  other  w^ords,  we  first  of  all  determine  what 
the  problem  dt.^iands,  times  or  parts,  then  operate  w^ith 
the  pure  number  symbols,  and  interpret  the  result  ac- 
cording to  the  conditions  of  the  problem. 

Illustrations  of  Division. — Let  us  take  a  few^  illus- 
trations of  these  inverse  operation^ :  (1)  We  count  out 
fifteen  oranges,  by  groups  of  five,  and  the  mimher  of 
groups  is  three.  We  count  them  out  in  five  groups,  and 
the  number  of  oranges  in  each  group  is  three.     These 


NATURE  OF  DIVISION  AND  FRACTIONS.        121 

are  said  to  be  two  totally  different  operations ;  for,  it  is 
alleged,  in  one  case  we  are  searching  for  the  size  (the 
numerical  value)  of  a  group — the  unit  of  measurement ; 
in  the  other  for  the  nnmher  of  groups.  But  a  little 
reflection  will  show  that  they  are  not  "  radically  differ- 
ent" operations;  they  are  psychological  correlates,  if 
not  identities.  In  counting  out  fifteen  oranges  in 
groups  of  five  there  is  a  count  of  ^'w,  then  another 
count  of  five,  then  another  count  of  five,  and  finally  a 
counting  of  the  number  of  groups.  Psychologically, 
in  counting  out  five  there  is  a  mental  sequence  of  five 
acts  (a  partial  synthesis) ;  this  is  repeated  three  times, 
and  finally  the  number  of  these  sequences  is  counted 
(complete  synthesis),  and  found  to  be  three.  In  the 
second  case,  where  the  number  {Jive)  of  groups  is  given, 
we  begin  by  putting  one  orange  in  each  of  five  places, 
making,  as  before,  a  *'  count "  of  Jive  oranges ;  this  opera- 
tion is  repeated  till  all  are  counted  out;  and  finally  we 
count  the  number  in  each  of  the  five  groups.  Tliat  is, 
there  is  a  mental  sequence  of  five  acts,  which  is  repeated 
three  times,  and  finally  the  number  of  such  sequences 
is  counted  in  counting  the  number  of  oranges  in  a  group. 
It  would  be  hardly  too  much  to  say  that  these  two  men- 
tal processes  are  so  closely  correlated  as  to  be  identical. 
Neither  the  three  times  in  the  one  question  nor  the 
three  oranges  in  the  other  can  be  found  without  count- 
ing out  the  whole  quantity  in  groups  of  five  oranges 
each  (see  page  75).  There  is  hardly  a  difference  even 
in  the  rhythm  of  the  mental  movement.  This  division 
by  counting  is  the  actual  process  w4th  things ;  it  is  the 
way  of  the  child  and  of  the  savage  or  the  illiterate  man  ; 
it  is  exactly  symbolized  in  the  "  two  kinds  of  division" 


I  1 


m 


Mi  I 


I    I 


ill:  I 


iM 


:      I 


i:    "I 


lit  i 


sli 

1, 


I 


^hl 


11 


122 


THE  PSYCHOLOGY  OF  NUMBER. 


— that  by  a  concrete  divisor  when  we  are  searching  for 
the  number  of  the  parts  as  actual  units  of  measure ;  and 
that  by  an  abstract  divisor  when  we  are  searching  for 
the  size  of  the  parts — i.  e.,  for  the  number  of  minor 
units  in  the  actual  unit  of  measure. 

With  this  actual  process  of  counting  out  the  objects 
the  arithmetical  operation  exactly  corresponds.  Work- 
ing by  long  division  as  more  typical  of  the  general 
arithmetical  operation,  we  have : 

I.  Division :  15  oranges  -r-  5  oranges  ;  i.  e.,  15 
oranges  are  to  be  counted  out  in  groups  of  5  oranges ; 
how  many  groups  ? 


5  oranges 


15 

oranges 

5 

10 

5 

5 

5 

1° — 1st  partial  multiplier 

l°_2d 

r_3d       «  " 


=  3  times. 


II.  Partition :  15  oranges  in  5  groups ;  how  many 
in  each  group  ? 


5  times 


15 

oranges 

5 

10 

5 

5 

5 

1  orange — 1st  partial  multiplicand  " 
1       "     —2d       "  " 


—3d 


« 


« 


=  3 

oranges. 


That  is,  once  more,  both  problems  are  solved  by  count- 
ing out  the  whole  quantity  in  groups  of  five. 

(2)  Solve  the  following  problems :  {a)  Find  the  cost 
of  a  town  lot  of  36  feet  frontage  at  $54  a  foot.  (J)  At 
the  rate  of  $54  a  foot,  a  town  lot  was  sold  for  $1944, 


NATURE   OF   DIVISION  AND  FRACTIONS.        123 


find  the  number  of  feet  fronta<^e.     (c)  Find  the  price 
per  foot  frontage  when  36  feet  cost  $1944. 


(a) 


i) 


$54 
36 


1620  =  30  times  $54 
324=    6     " 


$1944  =  36 
Or,  by  the  correlate,  $36  x  54 : 

(u)         $36 

54 

1800  =  50  times  $36 
_144=   4     " 
$1944  =  54     " 


$54)$1944 

1620  =  30  times  $54 
324=    6     " 
324 ;    36  times  $54 

(c)  Partition. 

36)$1944 

1800  =  $50  36  times 

144  =    $4  36     '' 

144 ;     $54  36  times 


On  comparing  the  successive  steps  in  {b)  with  those 
in  {a)  tliey  will  be  seen  to  correspond  exactly — that  is, 
(b)  is  the  exact  inverse  of  {ci).  But  the  steps  in  {c)  do 
not  correspond  with  those  in  (a),  the  operation  is  not 
the  exact  inverse  of  {a) ;  it  is  seen  to  be  the  exact  inverse 
of  the  correlative  {ii)  of  {ct).  This  indicates  the  con- 
nection of  the  operations  through  the  law^  of  commuta- 
tion ;  and  shows,  once  more,  that  either  of  the  corre- 
lated measurements  (i)  and  ((it)  may  be  used  in  the  solu- 
tion of  {c).  It  should  be  noted,  further,  that  {c)  is  a  case 
of  so-called  partition^  yet  involves  a  series  of  subtrac- 
tions that  is  a  series  of  partial  dividends  (why  not  par- 
tlendsf)  and  partial  quotients. 

N'ot  Two  Kinds  of  Division. — From  the  foregoing 
we  see  that  just  as  a  product  of  two  factors  may  be  in- 
terpreted in  two  ways,  so  there  may  be  two  interpre- 
tations of  the  result  of  the  inverse  operation,  division. 
The  factor  sought  may  be  either  the  numerical  value  of 
the  dividing  part  ("  derived  unit ")  in  terms  of  the  pri- 


I  h' 


e   i< 


I  '1 


II 


If 


124 


THE   PSYCHOLOGY   OF  NUMBER. 


ir't 


IT    I 


i 


:'     i 


'■\l 


I!; 


1.'     V   i 


..!,  f 


mary  luiit  which  measures  it,  or  the  numerical  vahie  of 
the  quantity  in  terms  of  the  derived  unit.  But  these 
numerical  values  may  be  interchanged  at  convenience, 
provided  the  results  are  rightly  interpreted.  There  are 
not  two  kinds  of  division  ;  there  is  one  operation  lead- 
ing to  one  numerical  result  having  two  related  mean- 
ings. It  seems  therefore  unnecessary,  either  on  psy- 
chological or  practical  grounds,  to  institute  two  kinds 
of  division — viz.,  division  (why  not  quot'dhmf)  in  the 
ordinary  sense  of  the  word,  and  "  partition  " — when  the 
search  is  for  the  numerical  value  of  the  measuring  quan- 
tity. When  the  search  is  for  the  mimerical  vahie  of  the 
measuring  unit,  is  not  the  pupil  likely  to  become  per- 
plexed by  a  series  of  parallel  definitions — of  divisor, 
dividend,  quotient — for  the  two  divisions  when  he  finds 
that  the  operations  in  both  cases  are  exactly  alike  ?  If 
there  is  confusion  in  using  the  term  division  in  two 
senses,  is  there  not  more  confusion  in  using  the  two 
terms,  divisor  and  quotient,  each  \\\t\\  two  different 
meanings  ?  Without  doubt,  the  meaning  of  the  result 
should  be  grasped  ;  but  this  can  not  be  done  by  simjfiy 
giving  two  names  to  exactly  the  same  arithmetical  opera- 
tion. Better  give  one  name  to  one  operation  resulting 
in  two  correlated  meanings  than  to  have  two  names  for 
one  and  the  same  operation.  The  new  name  does  not 
help  the  pupil  either  in  the  numerical  work  or  in  the 
interpretation  of  the  result.  How  is  the  child  to  know 
whether  a  given  problem  is  a  case  of  division  or  of 
"  partition  "  ?  He  can  not  know  without  an  intellectual 
operation,  analysis,  by  which  he  grasps  what  is  given 
and  what  is  wanted  in  the  problem.  In  other  words, 
he  must  know  the  meaning  of  the  problem,  must  know 


NATURE  OF   DIVISION   AND   FRACTIONS.        125 

whether  it  is  times  or  measuring  parts  he  is  to  search 
for,  before  he  heglns  the  operation  ^  to  this  knowledge 
the  different  names  afford  him  no  aid  wliatever.''^ 

Partition^  like  Division^  depends  on  Suhtractions. 
— It  is  said,  indeed,  that  in  "  jmrtition ''  we  are  search- 
ing for  the  numerical  value  of  one  of  a  given  number 
of  equal  parts  which  measure  a  quantity,  and  as  a  num- 
ber can  not  be  subtracted  from  a  measured  quantity, 
the  problem  can  not  be  solved  by  division.  To  this  the 
answer  is  easy :  In  the  first  place,  the  divisor  in  the 
arithmetical  operation  can  be  a  number,  and  the  sub- 
tractions rationally  explained  (see  page  122).  And,  be- 
sides, we  can  by  the  law  of  commutation  concrete  the 
number,  find  the  related  factor,  and  properly  interpret 
the  result.  But,  in  the  second  place,  if  the  divisor  can 
not  be  an  abstract  number,  what  magic  is  there  in  a 
strange  name  to  bring  the  impossible  within  the  easy 
reach  of  childhood  ?  It  seems,  according  to  the  parti- 
tionists^ that  20  feet  -i-  5  feet  represents  a  possible  and 
intelligible  operation  ;  but  that  20  feet  -r-  5  becomes 
possible  and  intelligible  only  by  calling  the  implied 
operation  a  case  of  "  partition  " ;  it  is  then  simply  one 
fifth  of  20  feet — that  is,  4  feet.  Certainly,  if  we  know  the 
multiplication  table,  we  know  that  one  fifth  of  20  feet  is 
4  feet,  but  we  know"  equally  well  that  5  feet  is  one  fourth 
of  20  feet.     These  are  not  typical  cases  for  the  ai-gu- 


*  Owing  to  the  fixed  unit  fallacy,  the  theory  of  the  '•  two  divi- 
sions" makes  an  unwarranted  distinction  between  the  actually  meas- 
uring part  and  its  times  of  repetition.  The  measuring  part,  as  well 
as  the  whole,  involves  both  the  spatial  element  (unit  of  quantity) 
and  the  abstract  (time)  element ;  it  is  itself  a  quantity  that  is  meas- 
ured by  a  minor  unit  taken  a  nrmber  of  times. 


'    I 


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HMli 


P 


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126 


TIIK   PSVCIIOLOUV  OF  NUMIJKR. 


I 


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m 


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inent ;  though  attention  to  the  processes  even  in  these 
cases  (see  page  122)  will  show  that  if  20  feet  -;-  5  is  impos- 
sible because  "  division  "  is  a  ])rocess  of  subtraction,  so 
also  is  the  process  one  fifth  of  20  feet,  because  "  pai'ti- 
tion  "  is  equally  a  process  of  sithtraction  (page  1 22).  For 
example,  the  operation  indicated  in  $ll-809f)28  -^  §4081 
it  is  admitted  involves  subtraction — i.  e.,  the  separation 
of  the  dividend  into  parts,  and  the  obtaining  of  partial 
quotients.  But  it  is  clear  that  l-4()81tli  of  this  dividend 
(partition)  is  obtained  by  exactly  the  same  2>vocess — i.  e., 
in  both  cases  we  have  a  first  sul)traction  of  3000  times 
the  divisor,  a  second  of  100  times,  a  third  of  80  times, 
and  a  fourth  of  3  times,  getting  the  same  numerical 
quotient  of  3183  in  both  operations ;  but  3183  is  inter- 
jpreted  as  pure  mi7nher  in  the  first  case,  and  as  measured 
guantity  in  the  second — the  so-called  partition. 

In  fine,  when  it  comes  to  pass  that  there  can  be  a 
clear  conception  of  a  foot  as  measured  by  inches  without 
the  thought  of  hoth  the  factors,  one  inch  and  tivelve  times^ 
then,  but  not  till  then,  it  may  be  rationally  aftirmed  that 
the  "two  divisions"  are  radically  different  and  totally 
unrelated  processes. 

Fractions. 

The  process  of  fractions  as  distinguished  from  that 
of  "  integers  "  simply  makes  explicit — especially  in  its 
notation — hoth  the  fundamental  processes^  division  and 
midti plication  {analysis-synthesis)^  which  are  involved 
in  all  numher. 

In  the  fundamental  psychical  process  which  con- 
stitutes number,  a  vague  whole  of  quantity  is  made 
definite   by  dividing  it  into  parts  and   counting  the 


[i:  r; 


NATURE  OF   DIVISION   AND   FRACTIONS.        127 

parts.  Tills  is  essentially  tlie  process  of  fractions.  Tlic 
"fraction,"  therefore,  involves  no  new  idea  ;  it  helps  to 
bring  more  clearly  into  consciousness  the  nature  of  the 
measuring  process,  and  to  express  it  in  more  definite 
form.  The  idea  of  ratio — the  essence  of  numher — is 
implied  in  simple  counting ;  it  is  more  definitely  used 
in  multiplication  and  division,  and  still  more  completely 
present  in  fractions,  which  use  both  these  operations. 
Fractions  are  not  to  be  regarded  as  something  different 
from  number — or  as  at  least  a  different  kind  of  number 
— arising  from  a  different  psychical  process  ;  they  are, 
in  fact,  as  just  said,  the  more  complete  development  of 
the  ideas  implied  in  all  stages  of  measurement.  So  far 
as  the  psychical  origin  of  number  is  concerned,  it  would 
be  more  correct  to  say  that  "  integers  "  come  from  frac- 
tions than  that  fractions  come  from  integers.  Without 
the  "  breaking  "  into  parts  and  the  "  counting  "  of  the 
parts  there  is  no  definitely  measured  Avhole,  and  no  ex- 
act nnmerical  ideas ;  the  definite  measurement  is  sim- 
ply (a)  the  number  of  the  parts  taken  distributively 
(the  analysis),  and  {h)  the  number  of  them  taken  col- 
lectively (the  synthesis).  The  process  of  forming  the 
integer,  or  whole,  is  a  process  of  taking  a  part  so  many 
times  to  get  a  complete  idea  of  the  quantity  to  be  meas- 
nred  ;  and  at  any  given  stage  of  this  operation  what  is 
reached  is  both  an  integer  and  a  fraction — an  integer  in 
reference  to  the  units  counted,  a  fraction  in  reference 
to  the  measured  unity. 

Even  in  the  imperfect  measurement  of  counting 
with  an  unmeasured  unit,  the  ideas  of  multiplication 
and  division  (and  therefore  of  ratio  and  fractions)  are 
implied   in   the  operation.     We  measure  a  whole  of 


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128 


THE  PSYCHOLOGY  OF  NUMBER. 


a  I 


fifteen  apples  by  threes  ;  we  count  the  parts — i.  e.,  re- 
late or  order  them  to  one  another,  and  to  the  \\'hole 
from  which  and  within  which  we  are  working.  This 
counting  has  a  double  reference — i.  e.,  to  the  unit  of 
measure,  and  to  the  whole  which  all  the  units  make  up. 
When,  for  example,  we  have  counted  two,  three,  .  .  . 
we  have  taken  one  unit  of  measure  two,  three,  .  .  . 
times,  and  each  count  is  expressed  or  measured  by  the 
numbers  two,  three,  .  .  .  — i.e.,  by  ''integers"  ;  but  also 
in  reaching  any  of  these  counts  we  have — in  reference 
to  tlie  whole — taken  one  of  the  five,  two  of  the  five, 
three  of  the  five,  etc. ;  that  is,  one  fifth  of  the  whole, 
two  fifths,  three  fifths,  etc. 

JVo  3feasure7nent  without  Fractions. — AYhen  we 
pass  to  measurement  with  exact  units  of  measure,  this 
idea  of  fractions — of  equal  parts  making  up  a  given 
whole — becomes  more  clearly  the  object  of  attention. 
The  conception,  3  aj^ples  out  of  5  apples  (three  fifths 
of  the  whole)  has  not  the  same  degree  of  clearness 
and  exactness  as  that  of  3  inches  out  of  a  measured 
whole  of  5  inches.  "VYhy  %  Because  in  the  former 
case  we  do  not  know  the  exact  value.,  the  how  much 
of  the  measuring  unit ;  in  the  latter  case  \\\q,  unit  is 
exactly  defined  in  terms  of  otlier  unities  larger  or 
smaller;  in  3  apples  the  units  are  alike ^  in  3  inches 
the  units  are  equal.  So  in  measuring  a  length  of  12 
feet  we  may  divide  it  into  2  parts,  or  3  parts,  or  4  parts, 
or  6  parts,  or  12  parts ;  then  we  can  not  really  think 
of  the  6  parts  as  making  the  whole  without  think- 
ing that  1  is  one  sixth  ;  2,  two  sixths ;  '  .  three  sixths, 
etc.  In  the  process  of  inexact  measurement  the  idea 
of  fractions  is  involved  ;  in  that  of  exact  measurement, 


•!  i 


NATURE  OF  DIVISION  AND  FRACTIONS.        I09 


I  f:^ 


I 


tliis  idea  is  more  clearly  defined  in  consciousness.  In 
short,  wherever  there  is  exact  measurement  there  is  the 
conception  of  fractions,  because  there  is  the  exact  idea 
of  number  as  the  instrument  of  measurement.  The 
process  of  fractions,  as  already  suggested,  simply  makes 
more  definite  the  idea  of  number,  and  the  notation  em- 
ployed gives  a  more  complete  statement  of  the  analysis- 
synthesis,  by  which  number  is  constituted.  The  num- 
ber Y,  for  example,  denotes  a  possible  measurement ; 
the  number  j^  states  more  definitely  the  actual  pro- 
cess. It  not  only  giv^es  the  absolute  number  of  units 
of  measure,  but  also  points  to  the  definition  of  the  unit 
of  measure  itself — that  is,  the  7  shows  the  absolute 
nmnber  of  units  in  the  quantity,  while  the  10  shows 
a  relation  of  the  unit  of  measure  to  some  other  stand- 
ard quantity,  a  primary  unit  of  reference  by  which  the 
actual  measuring  unit  is  defined.  If  a  quantity  is  di- 
vided into  2  equal  parts,  or  3  equal  parts,  or  4  equal 
parts,  or  71  equal  parts,  the  2,  3,  4,  .  .  .  n  shows  the 
entire  number  of  parts  in  each  measurement,  and  cor- 
responds with  the  *' denominator"  of  the  fraction  which 
expresses  the  measured  quantity  as  unity  y  nnd  in  count- 
ing up  (^-numerating)  the  parts  (units)  we  are  constantly 
making  "nnmerators" — e.  g.,  1  out  of  n,  2  out  of  n^ 

11 
3  out  of  /?,  etc.  :   or  1-nth,  2-wths,  .  .  .  ?i-nths,  or  - , 

■n 

which  is  the  measured  unity.  Or,  if  attention  is  given 
to  tlie  measuring  units — the  ones — the  parts  are  ex- 
pressed by  1,  2,  8,  etc.,  and  the  measured  quantity  itself 

n 
is  expressed  by  y.      Again,  measuring   the   side   of  a 

certain  room,  we  find  it  to  contain  ^  yards.     This  is  a 


(  1::^= 


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130 


TOE   PSYCHOLOGY   OF  NUMBER. 


full  statement  of  the  process  of  measurement ;  it  means 
(1)  that  the  primary  unit  of  measurement  (the  standard 
of  reference)  is  one  yard,  (2)  that  the  derived  unit  of 
measurement  is  one  third  of  this,  and  (3)  that  this  de- 
rived unit  is  taken  nineteen  times  to  measure  the  quan- 
tity. This  is  seen  to  agree  with  the  mental  process  of 
the  exact  stage  of  measurement  in  which  the  unit  of 
measure  is  itself  defined  or  measured  (see  page  94). 
There  must  be,  as  we  have  seen,  (1)  a  standard  unity 
of  reference  (the  primary  unit),  (2)  a  derived  unit  (the 
unit  of  direct  measurement),  and  (3)  the  number  of 
these  in  the  quantity.  The  fraction  gives  complete 
expression  to  this  process  :  In  $f ,  for  example,  (1)  the 
dollar  is  the  unit  of  reference  ;  (2)  it  is  divided  into  four 
parts  to  get  the  derived  unit — the  actual  unit  of  meas- 
ure ;  (3)  the  "  numerator  "  3  shows  how  many  of  these 
units  make  up  the  given  quantity,  and  expresses  the 
ratio  qf  this  quantity  to  the  standard  unity. 

So,  again,  the  measurement — 19  feet — of  the  side 
of  a  room  can  be  stated  in  tei-ms  of  other  units  of  the 
scale.  It  is  12  X  19  inches,  or  19  ~-  3  yards,  and  the 
first  of  these  expressions,  as  well  as  the  second,  is  one 
of  fractions ;  it  is  -^-f-S- — that  is,  not  228  ones  merely,  but 
228  of  a  definite  \mit  of  measure — namely,  one  twelfth 
of  a  foot ;  just  as  the  second  is  ^ — i.  e.,  19  times  a  unit 
of  measure  defined  by  its  relation  to  the  yard.  In  the 
former  case  we  do  not  generally  state  the  measurement 
in  fractional  form,  but  the  interj>retation  of  it  demands 
an  explicit  reference  to  a  dtmominator.  Note  what 
this  brings  us  to  :  19  (feet)  —  228  (inches)  =  ^^-  (yards) 
—  t¥"  (I'ods) — that  is,  four  entirely  different  numbers 
equal  to  one  another;   a   result  which   must  appear 


i 


s 


NATURE  OF  DIVISION  AND  FRACTIONS.        131 


I 


utterly  meaningless  to  a  child  who  has  been  trained  by 
the  fixed  unit  method.  Any  method  which  treats  num- 
ber as  a  name  for  physical  objects  can  not  but  reach  just 
such  absurdities.  Only  the  method  which  recognises 
that  number  is  a  psychical  process  of  valuation  (analy- 
sis-synthesis) is  free  from  such  difficulties.  The  unit 
does  not  designate  a  fixed  thing ;  it  designates  simply 
the  unit  of  valuation,  the  how  much  of  anything  which 
is  taken  as  one  in  measuring  the  value  (or  how  much) 
of  a  group  or  unity.  It  defines  how  many  units  each 
of  so  much  value  make  up  the  so  much  of  the  whole. 
The  complete  process  is  one  of  fractions,  and  the  full 
statement  of  it  is  a  fraction,  whether  written  out  in  full 
or  necessarily  understood  in  the  interpretation.  The 
228  inches  is  -S^,  signifying  that  the  number  of  the 
derived  units  of  measure  in  one  inch  is  1 ;  19  feet  is  ^^- 
yards,  signifying  that  the  number  of  the  derived  units 
of  measurement  in  one  yard  is  3 ;  the  ^^  rods  show 
that  the  number  of  units  of  measurement  in  one  rod  is 
11 ;  in  other  words,  the  unit  of  measure  in  ^-  is  one  of 
the  three  equal  parts  of  one  yard,  etc. 

It  appears,  therefore,  that  every  numerical  operation 
which  makes  a  vague  quantity  definite,  when  fully  stated, 
involves  the  "  terms  "  of  a  fraction — that  is,  a  fraction 
may  be  considered  as  a  convenient  language  (notation) 
for  expressing  quantity  in  terms  of  the  process  which 
measures  or  defines  it — which  makes  it  "  number." 

A  fraction,  then,  completely  defines  the  unity  of  ref- 
erence, and  thus  determines  the  unit  of  measure  for  the 
quantity  that  is  to  be  measured.  Thus  the  inch  may  be 
defined  from  |f  foot,  the  foot  from  }  yard,  the  ounce 
from  -J-J  pound,  the  cent  from  -J  J^  dollar,  etc.     In  each 


'      [M 


is 


fV 


132 


THE  PSYCHOLOGY  OP  NUMBER. 


I  ; 


in 


'    .1 


I ; 


1 


i  } 

"V 

lii;: 

li: 


I'l'' ; 


case  the  denominator  shows  the  analysis  of  a  standard 
unity  into  units  of  measurement — i.  e.,  the  unity  in 
terms  of  the  units  taken  collectively.  Thus  the  meas- 
urements of  the  quantities  Y  inches,  5  ounces,  35  cents 
are  more  explicitly  stated  by  the  respective  fractional 
forms  -^  foot,  -^  pound,  y^/^j-  dollar,  because  the  unit 
of  measure  in  each  case  is  consciously  defined  by  its  re- 
lation to  a  standard  unity  in  the  same  scale. 

It  is  clear  that  the  definition  of  number  (page  71) 
includes  the  fraction,  for  in  both  fraction  and  integer 
the  fundamental  conception  is  that  of  a  quantity  meas- 
ured by  a  number  of  defined  parts — the  conception  of 
the  ratio  of  the  quantity  to  the  measuring  unit.  The 
fraction  differs  from  the  integral  number — in  so  far  as 
it  differs  at  all — in  defining  the  measuring  unit,  and 
thus  giving  more  completely  the  psychical  operation  in 
the  exact  stage  of  measurement. 

If  the  fraction,  as  being  a  number,  is  a  mode  of 
measurement,  tliere  appears  to  be  no  need  of  a  special 
definition  of  it  as  the  foundation  of  a  new  or  different 
class  of  numerical  operations.  The  definitions  which 
ignore  fractions  as  a  mode  of  measurement  are  in  gen- 
eral vague  and  inaccurate,  and  lead  to  much  perplexity 
in  the  treatment  of  fractions.  It  is  hardly  accurate  to 
gay  tliat  a  ''  fraction  is  a  number  of  the  equal  parts  of  a 
unit,"  or  that  "  it  originates  in  the  division  of  the  unit 
into  equal  parts."  Here  the  important  distinction  be- 
tween unity  and  a  unit  is  overlooked.  Measuring  a 
piece  of  cloth  we  find  it  contains  four  yards  :  before  meas- 
urement it  was  mere  unity,  after  measurement  a  defined 
unity  ;  but  in  neither  case  is  it  a  iintt.  It  is,  after  meas- 
urement, a  unity  of  U7i{ts — a  sum.     Is  or  is  it  entirely 


u 


NATURE  OF  DIVISION  AND  FRACTIONS.        133 


consistent  with  the  measuring  idea  to  say  that  a  frac- 
tion is  one  or  more  of  the  equal  parts  of  a  unity.  Of 
course,  in  counting  the  equal  parts  of  a  measured  whole 
— a  unity — we  take  a  number  of  parts  in  making  the 
synthesis  of  all  the  parts.  But  since  a  fraction  is  a 
number,  and  therefore  denotes  measured  quantity,  it 
denotes  a  whole  quantity,  a  unity — e.  g.,  -J  of  a  yard  is 
as  much  a  quantity — a  measured  xinity — as  4  yards  or 
40  yards  ;  it  is  a  fraction  in  its  relation  to  a  larger 
unity,  the  yard  taken  as  a  standard  of  reference. 

The  Improper  Fraction. — From  the  same  misappre- 
hension of  the  nature  of  number  endless  discussions 
arise  regarding  the  classes  of  fractions  "  proper,"  "  im- 
proper," etc.  With  a  right  conception  of  the  meas- 
uring function  of  a  fraction  there  is  no  mystery  about 
the  "  improper "  fraction.  From  the  definition  of  a 
fraction  as  a  "  number  of  the  equal  parts  of  a  unit,"  it 
is  inferred,  e.  g.,  that  -f  of  a  yard  can  not  be  a  fraction, 
because  it  represents  not  parts  of  a  unit,  but  the  whole 
unit  and  something  more.  Since  3  thirds  make  up  the 
yard  (the  unit),  whence  come  the  4  thirds  ? 

In  this  objection  we  have  the  fallacy  of  the  fixed 
unit  as  well  as  the  misapprehension  of  the  nature  of 
number.  The  fraction  in  the  expression  |-  of  a  yard  is 
a  number.  It  means  the  repetition  of  a  unit  of  meas- 
ure to  equal  a  certain  quantity.  This  unit  of  measure 
is  not  the  yard  ;  it  is  a  unit  defined  by  its  relation  to 
the  yard  ;  it  is  one  of  the  three  equal  parts  into  which 
the  unity  yard  is  divided  to  get  the  direct  unit  of  meas- 
ure ;  and  there  is  absolutely  nothing  to  make  the  yard 
the  limit  of  quantity  to  which  this  unit  can  be  applied. 
The  yard  is  the  primary  unit  of  reference  from  which 


■■%] 


"  '.II 


m 


134 


THE  PSYCHOLOGY  OF  NUMBER. 


i'[ 


m 


ill' 


I!.:., 


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IM!' 


•  r 


the  actual  measuring  unit  is  derived,  and  there  is  no 
more  mystery  in  the  application  of  this  unit  to  measure 
a  quantity  greater  than  the  primary  unit  than  in  the 
measured  quantity,  3  feet  x  4,  because  it  is  greater 
than  one  foot^  the  primary  unit  from  which  the  meas- 
uring unit  (3  feet)  is  derived. 

The  expression  4  thirds  of  a  yard  indicates  an 
exactly  measured  quantity ;  exactly  measured,  because 
the  unit  of  measure  is  itself  measured  in  its  relation 
to  another  quantity  of  the  same  scale.  This  properly 
defined  unit  (1  third  of  a  yard)  can  be  applied  to  any 
homogeneous  quantity  whatever,  and  may  be  contained 
in  such  quantity  one,  two,  three,  four,  ,  ,  .  n  times ;  in 
fact,  4  thirds  yard,  5  thirds  yard,  .  ,  .  n  thirds  yards 
are  only  different  and  more  exact  ways  of  stating  the 
measurements — 4  feet,  5  feet,  .  .  .  n  feet. 

The  Compound  Fraction. — Nor  is  there  any  dif- 
ficulty in  interpreting  a  "  compound  "  fraction.  The 
value  of  8  yards  of  cloth  at  %\  a  yard  is  expressed  by 
$|-  X  8,  a  measurement  which  ought  to  occasion  no 
more  perplexity  than  $3  x  8,  when  it  is  understood 
that  the  denominator  merely  defines  the  unit  of  meas- 
ure with  reference  to  the  primary  unit.  So  the  value 
of  f  yard  of  cloth  at  $8  a  yard  is  expressed  by  $8  x  J, 
a  measured  quantity  where,  once  more,  the  denominator 
shows  how  the  unit  of  measure  is  to  be  obtained — i.  e., 
it  shows  which  of  the  myriad  ways  of  parting  and 
wholing  $8 — the  unity  of  reference — will  give  the  direct 
or  absolute  unit  of  measure.  This  explanation  applies 
to  %-^-i  X  J,  and  to  any  compound  fraction  whatever. 

The  Complex  Fraction. — It  is  said  that  the  complex 
fraction  is  an  impossibility,  because  a  quantity  can  not 


NATURE  OF  DIVISION  AND  FRACTIONS.        135 


be  divided  into  a  fractional  number  of  equal  parts — e.  g., 
if  tlie  denominator  of  such  a  fraction  is  J,  it  implies  the 
division  of  some  unity  into  3  fourths  equal  parts,  which 
is  absurd.  This  is  to  restrict  the  term  fraction  by  the 
imperfect  definition  already  quoted,  which  ignores  num- 
ber as  measurement  and  fractions  as  an  explicit  state- 
ment of  the  measuring  process.  Division  and  multi- 
plication are  fundamental  in  the  psychical  process  of 
defining  quantity  ;  the  fraction  simply  brings  the  pro- 
cess articulately  into  consciousness,  and  by  its  notation 
gives  it  complete  expression.  The  statement  that  the 
fraction  process  and  the  division  process  are  totally  dis- 
tinct is  so  far  from  being  true,  that  there  is  no  division 
without  the  fraction  idea,  and  no  fraction  without  the 
division  idea.  Both  are  identified  by  the  law  of  com- 
mutation— a  law  which  is  the  expression  of  a  necessary 
and  universal  action  of  the  mind  in  the  measurement  of 
quantity.  The  symbol  f  foot  is  an  exact  expression  for 
a  measured  quantity  ;  like  every  other  such  expression, 
it  defines  the  unit  of  measure  and  denotes  the  times  this 
unit  is  repeated  ;  and,  like  every  such  expression,  it  has 
two  interpretations  corresponding  to  the  related  concep- 
tions of  the  measured  quantity  :  it  is  i  foot  x  3,  or  3 
feet  X  J.  We  shall  be  justified  in  treating  these  two 
things  (the  fractic-n  and  division)  as  entirely  distinct 
when  we  are  able  to  conceive  that  3  feet  x  4,  and  4 
feet  X  3  are  unrelated  measurements  of  totally  different 
quantities. 

It  may  be  noted,  then,  that  in  the  "  complex  "  frac- 
tion just  as  in  division  there  may  be  two  interpretations. 
In  $12  -s-  $4  the  measuring  is  not  hyfour  parts — it  is  a 
parting  hy  fours  ;  while  in  $12  -j-  4  there  is  a  measur- 


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THK  rSYCHOLOGY  OF  NUMBER. 


ing  by  four  parts — it  is  a  parting  by  threes.  But  in 
every  case,  as  has  often  been  shown,  either  the  size  of 
tlie  parts  is  given  to  iind  their  number,  or  the  number 
of  the  parts  to  find  their  size.     The  same  thing  holds  in 

so-called  complex  fractions.     In  ^  — as,  e.  g.,  find  how 

much  cloth  at  $J  a  yard  can  be  bought  for  $9 — it  is  not 
proposed  to  divide  $9  into  $J  equal  parts,  but  to  find 
the  times  the  measuring  unit  ^J  is  taken  to  make  $9. 

N'or  in     ;o  — as,  e.  g.,  find  cost  per  yard  ^vhen  $9  was 

paid  for  J  yard — is  there  any  attempt  to  divide  $9 
into  3  fourths  equal  parts.  The  purpose  is  to  find  the 
quantity  which,  with  J  as  multiplier  (i.  e.,  taken  J  times), 
will  give  $9 ;  and  it  is  a  matter  of  indifference  whether 
the  expression  is  called  a  "complex"  fraction  or  di- 
vision of  fractions,  for  fractions  are  necessarily  corre- 
lated with  multiplication  a7id  division  by  the  uniform 
action  of  the  mind  in  dealing  w^ith  quantity. 

Summary  and  Applications. 

1.  All  numerical  operations  are  intrinsically  con- 
nected with  number  as  measurement,  and  distinguished 
from  one  another  through  the  development  of  the 
measuring  idea  in  psychological  complexity.  Addi- 
tion and  subtraction  have  their  origin  in  the  oper- 
ation of  counting  with  an  unmeasured  unit — they  do 
not  explicitly  use  the  idea  of  ratio,  but  merely  that  of 
more  or  less — the  idea  of  aggregation.  Multiplication 
and  division  have  their  origin  in  the  use  of  an  exact 
unit  of  measure — a  unit  w'hich  is  itself  defined — and, 


NATURE  OF  DIVISION  AND  FRACTIONS.        137 


■I     \i\ 


besides  tlie  idea  of  aggregation,  use  the  idea  of  ratio. 
It  follows,  accordingly,  that  addition  and  subtraction 
should  precede  in  order  of  formal  instruction,  inulti})li- 
cation,  and  division.  Addition  and  subtraction,  being 
inverse  operations,  should  go  together,  with  the  em- 
2))jasis  at  first  slightly  upon  addition. 

2.  Multiplication  and  division,  being  inverse  oper- 
ations, should  go  together,  with  the  emphasis  first  upon 
multiplication.  Multiplication  should  not  be  taught  as  a 
case  of  addition,  nor  division  as  a  case  of  subtraction. 
But  the  factor  idea  (ratio  or  number)  should  in  each 
case  displace  the  idea  of  aggregation.  While  this  is  the 
order  of  analytical  instruction,  the  processes  involved 
in  multiplication  should  be  used — that  is,  in  primary 
teaching  there  should  be  frequent  excursions  into  these 
processes  in  accordance  with  the  fundamental  psycho- 
logical law  :  "  First  the  rational  use  of  the  process,  and 
ultimately  conscious  recognition  of  it." 

3.  In  multiplication  the  multiplicand,  strictly  speak- 
ing, always  represents  a  measured  quantity,  and  is  com- 
monly said  to  be  "concrete";  the  multiplier  always 
represents  pure  number — the  ratio,  in  fact,  of  the  prod- 
uct to  the  multiplicand.  But,  as  the  multiplicand  always 
involves  the  idea  of  number  (it  expresses  the  mimher  of 
primary  measuring  units),  the  two  factors  of  the  product 
may  be  interchanged — that  is,  the  multiplier  may  be  made 
the  concrete  quantity,  and  the  multiplicand  the  pure  num- 
ber denoting  times  of  repetition. 

4.  Division  is  the  inverse  of  multiplication.  We 
have  the  product  given  and  one  of  the  two  factors 
which  produce  it  to  find  the  other  factor.  And  since 
there  are  two  interpretations  of  the  process  of  multipli- 


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THE  PSYCHOLOGY  OF  NUMBER. 


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cation  there  may  be  two  interpretations  of  its  inverse 
process,  division.  But  there  are  not  two  kinds  of  mul- 
tiplication nor  two  kinds  of  division.  In  each  case 
there  is  one  process  and  one  result  with  two  interpreta- 
tions. No  assistance  to  this  interpretation  can  be  af- 
forded by  giving  the  name  "  partition  "  to  the  process 
by  which  we  find  the  size,  or  numerical  value,  of  the 
measuring  part.  The  student  knows  what  he  is  look- 
ing for  before  he  begins  the  operation — whether  for  the 
value  of  the  parts  in  terms  of  the  primary  unit,  or  the 
number  of  them  in  the  whole  quantity.  It  is  not  neces- 
sary for  him  to  give  a  new  name  to  an  old  operation. 
Besides,  if  he  does  not  know  what  he  is  searching  for 
before  he  begins  the  numerical  work,  the  new  name 
throws  no  light  upon  the  subject. 

From  the  relation  existing  between  nmltiplication 
and  division,  it  is  seen  that  in  division  the  dividend — or 
multiplicand,  as  being  the  product  of  two  factors — always 
represents  a  measured  quantity — i.  e.,  it  is  concrete  ;  the 
divisor  may  denote  either  a  concrete  quantity  or  a  pure 
number ;  and  the  quotient  is  of  course  numerical  in  the 
one  case  and  interpreted  as  concrete  in  the  other. 

5.  In  fractions  there  are  no  mental  processes  differ- 
ent from  what  are  involved  in  number  as  a  mode  of 
measuring  quantity.  The  psychical  process  by  which 
number  is  formed  is  from  first  to  last  essentially  a  pro- 
cess of  "  fractioning  " — making  a  whole  into  equal  parts 
and  remaking  the  whole  from  the  parts.  In  the  pro- 
cess of  number  we  start  with  a  whole ;  we  have  a  unit  of 
measurement;  we  repeat  the  unit  of  measurement  to 
make  up  the  whole.  In  a  measured  quantity  repre- 
sented by  a  fraction  we  do  exactly  the  same  thing.    We 


1*1 


1 


NATURE  OF  DIVISION  AND  FRACTIONS.        130 


begin  with  a  whole  of  quantity  ;  we  use  a  unit  of  meas- 
ure of  the  same  kind  as  the  quantity;  we  repeat  the 
unit  of  measure  to  make  up  the  whole.  The  fraction 
by  its  notation  brings  out  more  explicitly  the  actual  pro- 
cess of  measurement — that  is,  it  not  only  gives  the  num- 
ber of  units  of  measure,  but  actually  defines  the  unit 
itself  in  terms  of  some  other  unit  in  the  same  scale.  In 
other  words,  a  fraction  sums  ujp  in  one  statement  the 
mental  pr^ocess  of  analysis-synthesis  Jjy  which  a  vague 
whole  is  made  definite. 

1.  As  fraction  at  all  it  expresses  a  portion  of  some 
group  or  whole  with  which  the  quantity  represented 
by  the  fraction  is  compared,  and  wdiich  defines  the 
measuring  unit.  Thus,  |-  yard  of  cloth  is  itself  a 
whole,  a  definitely  measured  quantity ;  but  it  is  a  frac- 
tion as  regards  the  standard  of  reference,  yard.,  which 
defines  the  direct  measuring  unit,  one  eighth  of  a  yard. 

2.  A  fraction,  therefore,  always  denotes  {a)  the  ab- 
solute number  of  units  in  a  measured  quantity ;  {Jj)  the 
number  of  such  units  in  some  standard  quantity  which 
defines  the  measuring  unit  in  {a) ;  and  {c)  the  ratio  of 
the  given  quantity  (represented  by  the  fraction)  to  this 
standard  of  reference.  The  numerator  of  the  fraction 
gives  {a)  and  {c\  and  the  denominator  gives  (b).  Of 
course,  any  part  (or  multiple)  of  the  standard  of  refer- 
ence may  be  taken  as  the  unit  of  measure  for  a  given 
quantity ;  a  given  length  may  be  measured  by  1  foot, 

or  by  1000  feet,  or  by  -r^Vir  ^^  ^  f^*^*-  ^^^  ^^  begin- 
ning the  explicit  treatment  of  fractions  it  is  better  to 
use  certain  standard  measures,  their  subdivisions,  and 
their  relations  to  one  another.  Thus,  as  a  process  of 
analysis-synthesis,  the  foot  is  defined  by  -J-f,  the  yard 


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140 


THE  PSYCHOLOGY   OF   NUMHKR. 


by  f,  the  pound  by  fj,  the  dollar  by  -fj  or  by  fj>^; 
where  12  refers  to  inches,  3  to  feet,  KJ  to  ounces,  10  to 
dimes,  and  100  to  cents.  Familiarity  with  fractions 
thus  defined  by  and  connected  with  the  ordinary  scales 
of  measurement  means  easy  mastery  of  all  forms  of 
fractions  as  a  mode  of  definite  measurement. 

3.  As  to  the  teaching  of  fractions,  it  will  be  enough, 
for  the  present,  to  note  the  following  points : 

1.  In  the  formal  treatment  of  fractions  nothing  new 
is  involved  ;  there  is  simply  a  eonscums  direction  of  at- 
tention to  ideas  and  processes  which,  under  right  teach- 
ing, have  been  used  from  the  first  in  the  formation  of 
numerical  ideas,  and  which  have  been  further  developed 
in  the  fundamental  arithmetical  operations. 

2.  As  in  "integers"  so  in  teaching  fractions,  the 
idea  and  process  of  measurement  should  be  ever  present. 
To  begin  the  teaching  of  fractions  with  vague  and  un- 
defined "units"  obtained  by  breaking  up  equally  unde- 
fined wholes— the  apple,  the  orange,  the  piece  of  paper, 
the  pie — may  be  justly  termed  an  irrational  procedure. 
Half  a  pie,  e.  g.,  is  not  a  numeral  expression  at  all,  un- 
less the  pie  is  defined  by  weight  or  volume  ;  the  con- 
stituent factors  of  a  fraction  are  not  present ;  the  unity 
of  arithmetic  is  ignored ;  the  process  of  fractions  is 
assumed  to  be  something  different  from  that  of  num- 
ber as  measurement ;  it  becomes  a  question — it  ac 
ally  has  been  questioned — whether  a  fraction  is  really  ,i 
number;  and  all  this  in  spite  of  the  fact  that  from  the 
beginning  fractions  are  implicit  in  all  operations;  that 
from  first  to  last  the  process  of  number  as  a  psychical 
act  is  a  process  of  fractions. 

3.  The  primary  step  in  the  explicit  teaching  of  frac- 


NATURE  OP    DIVISION   AND   FRACTIONS.        141 

tioiis — that  is,  in  making  tlie  liabit  of  fractioning  already 
formed  an  object  of  analytical  attention — is  to  make 
perfectly  definite  the  child's  acquaintance  with  certain 
standard  measures,  their  subdivisions  and  relations.  In 
all  fractions — because  in  all  exact  measurement — there 
must  be  a  definite  imit  of  measure.  This  implies  two 
things  :  {a)  The  definition  of  a  standard  of  reference 
(the  "  primary  "  unit)  in  terms  of  its  own  unit  of  meas- 
ure ;  ih)  the  measurement  of  the  given  quantity  by 
means  of  this  "  derived  "  unit.  If  the  foot  is  unit  of 
measure,  it  is  unmeaning  in  itself ;  it  must  be  mastered, 
must  be  given  significance  by  relating  it  to  other  units 
in  the  scale  of  length ;  it  is  1  (yard)  -^-  3  in  one  direc- 
tion ;  or  (taking  the  usual  divisions  of  the  scale)  it  is 
If  (i.  e.,  -^  X  1^)  in  the  other  direction,  i.  e.,  as  meas- 
ured in  inches.  The  teaching  of  fractions,  then,  should 
be  based  on  the  ordinary  standard  scales  of  measure- 
ment ;  on  the  fundamental  process  of  parting  and  whol- 
ing  in  measurement,  and  not  upon  the  qualitative  parts 
of  an  undefined  unity. 

4.  Under  proper  teaching  of  number  as  measure- 
ment the  pupil  soon  learns  to  identify  instantly  4  inches, 
-^  foot,  -J  foot  as  expressions  for  the  same  measured 
quantity.  He  is  led  easily  to  the  conscious  recognition 
of  the  true  meaning  of  fractions  as  a  means  of  indicating 
the  exact  measurement  of  a  quantity  in  terms  of  a  meas- 
uring unit  which  is  itself  exactly  measured. 

5.  Addition  and  subtraction  of  fractions  involve  the 
]    'nciple  of  ratio,  multiplication  and  division  the  prin- 

'le  of  proportion.     In  all  cases  the  meaning  of  frac- 

v)ns  as  denoting  definitely  measured  quantity  should 

i/e  made  clear.     For  example,  not  i  x  J,  but  J  foot  x  j ; 


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TIIH   PSYCHOLOGY  OF  NUMBER. 


not  f  X  I",  but  $f  X  J,  as  indicating,  e.  g.,  the  cost  of  -J 
yard  of  cloth  at  $£  a  yard. 

Since  a  fraction  expresses  a  quantity  in  a  form  for 
comparison  with  other  quantities  of  the  same  kind,  the 
fundamental  operations  as  applied  in  fractions  carry  out 
these  comparisons.  Addition  is  always  (as  in  "  whole 
numbers  ")  of  homogeneous  quantities — i.  e.,  thr.jje  meas- 
ured in  terms  of  some  unit  of  length,  surface,  volume, 
time,  etc. ;  so  wdth  subtraction.  All  the  first  examples 
should  deal  only  with  definite  measures ;  after  the  prin- 
ciple is  quite  familiar,  and  only  then,  fractions  having 
denominators  not  corresponding  to  any  existing  scale 
of  measurement — e.  g.,  IT,  49,  131 — may  be  introduced 
for  the  sake  of  securing  mechanical  facility. 

The  same  remark  applies  to  multiplication  and  divi- 
sion of  fractions — opcations  wdiich  involve  no  princi- 
ples different  from  the,  corresponding  operations  wath 
"  whole  numbers."  Multiplication  of  fractions  is  7nul- 
tipUcation,  and  division  is  divisio7i  /  they  are  not  new 
processes  under  old  names.  They  make  explicit  use  of 
ratio  (the  comparison  of  quantities),  which  is  implied  in 
the  operations  with  "  integers,"  by  defining  the  measur- 
ing unit  whicl^  defines  a  measured  quantity.  They  put 
in  shorthand,  as  it  \vere,  the  complete  psychical  process 
of  measurement,  and  thus  make  a  severer  demand  on 
conscious  attention.  But  if  number  has  been  from  the 
first  taught  upon  the  psychological  method,  the  pupil 
will  be  quite  prepared  to  meet  this  demand.  Tliere 
will  be  nothing  strange  in  reducing  ^^ractions  to  a  com- 
mon denominator,  nor  any  mystery  in  a  product  less 
than  the  multiplicand,  or  in  a  quotient  greater  than  the 
dividend ;  so  far  as  the  nature  of  the  processes  is  con- 


I 


NATURE  OF  DIVISION  AND  FRACTIONS.        I43 


:  ined,  $-|-  +  $J  will  be  just  as  intelligible  as  $3  +  §i. 
If,  too,  the  nature  and  relation  of  twies  and  nieasurir.g 
jparts  have  become  fannliar,  there  will  be  no  more 
mystery  in  18  feet  x  j-  =  9  feet  than  in  measuring  half- 
way across  a  room  18  feet  wide ;  the  peculiar  thing 
would  be  if  taking  a  quantity  only  a  part  of  a  time  did 
not  give  a  smaller  quantity. 

So  in  division,  when  the  mutual  relati«  i  between 
times  and  parts  ig  understood,  the  operation  %^  -r-  ?^yV) 
or  $f -r- xV>  ^^  j^'^t  ^^s  intelligible  as  $80 -r- ^10,  or  us 
$80  -f- 10.  To  say  that  the  quotient  e'ujld^  the  result 
of  $-f  -^  $ii)-)  is  greater  than  the  dividend  ($f )  is  to  talk 
nonsense  ;  is  to  compare  incomparable  tilings — is  to 
confuse  parts  with  times,  quantity  with  number,  matter 
wdth  a  psychical  process. 


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CHAPTER  YIII. 

ON    PRIirARY    NUMBER   TEACHING. 

The  Ntimler  Instinct. — AYe  liave  seen  that  number 
is  not  something  impressed  upon  the  mind  by  external 
energies,  or  given  in  the  inere  perception  of  things,  but 
is  a  product  of  the  mind's  action  in  the  measurement 
of  quantity — that  is,  in  making  a  vague  whole  definite. 
Since  this  action  is  the  fundamental  psychical  activity 
directed  upon  quantitative  relations,  the  process  of  num- 
bering should  be  attended  with  interest;  that  is,  con- 
trary to  the  connnoidy  received  opinion,  the  study  of 
arithmetic  should  be  as  interestino;  to  the  learner  as  that 
of  any  other  subject  in  the  curriculum.  The  training 
of  observation  and  perception  in  dealing  with  nature 
studies  is  said  to  be  universally  interesting.  This  is  no 
doubt  true,  as  there  is  a  hunger  of  the  senses — of  sight, 
hearing,  touch — which,  when  gratified  by  the  j^resenta- 
tion  of  sense  materials,  affords  satisfaction  to  the  self. 
But  we  may  surely  say  with  equal  ti'uth  that  the  exer- 
cise of  the  higher  energy  which  works  upon  these  raw 
materials  is  attended  ^nth  at  least  e(jual  pleasure.  The 
natural  action  of  attention  and  judgment  working  upon 
the  sense-facts  must  be  accompanied  with  as  deep  and 
vivid  an  interest  as  the  normal  action  of  the  observing 
powers  through  which  the  sense-facts  are  acquired. 

144 


Kt'l 


!l 


number 
external 
ngs,  but 
uremcnt 
definite, 
activity 
i  of  niun- 
;  is,  con- 
study  of 
as  that 
traininoj 
li  nature 
lis  is  no 
of  sight, 
I'csenta- 
thc  self, 
he  exer- 
lese  raw 
•e.     The 
ng  upon 
Jeep  and 
hserving 
red. 


ON  PRIMARY  NUMBER  TEACHING. 


145 


For  numerical  ideas  involve  the  simplest  forms  of 
this  higher  process  of  mental  elaboration  ;  they  enter 
into  all  human  activity  ;  they  are  essential  to  the  proper 
intei'pretation  of  the  pliysical  world  ;  they  are  a  neces- 
sary condition  of  man's  emancipation  from  the  merely 
sensuous  ;  they  are  a  powerful  instrument  in  his  reac- 
tion against  his  environment ;  in  a  word,  number  and 
numerical  ideas  are  an  indispensable  condition  of  the 
development  of  the  individual  and  the  progress  of  the 
race.  It  would  therefore  seem  to  be  contrary  to  the 
"beautiful  economy  of  Nature"  if  the  mind  had  to  be 
forced  to  the  acquisition  of  that  knowledge  and  power 
which  are  essential  to  individual  and  racial  develop- 
ment ;  in  other  words,  if  the  conditions  of  progress 
involved  other  conditions  which  tended  to  retard 
})rogress. 

The  position  here  taken  on  theoretical  grounds,  that 
the  normal  activitv  of  the  mind  in  constructinc:  number 
is  full  of  interest,  is  confirmed  by  actual  experience  and 
observation  of  the  facts  in  child  life.  There  are  but 
few  children  who  do  not  at  first  deliijjht  in  number. 
Counting  (the  fundamental  process  of  arithmetic)  is  a 
thing  of  joy  to  them.  It  is  the  promise  and  potency  of 
liigher  things.  The  one,  two,  three  of  the  "  six-years' 
darling  of  a  pygmy  size  "  is  the  expression  of  a  higher 
energy  struggling  for  complete  utterance.  It  is  a  proof 
of  his  gradn'^l  emergence  from  a  merely  sensuous  state 
to  that  hiffher  staffe  in  which  he  bcinns  to  assert  his 
mastery  over  the  physical  world.  We  have  seen  a  first- 
year  class — the  whole  class — just  out  of  the  kindergar- 
ten, become  so  thoroughly  interested  in  arithmetic  under 
a  sympathetic  and  competent  teacher,  as  to  i)refev  an 


I  • 


\ii 


'I        r^A 


< 


vm 


146 


THE  PSYCnOLOGY   OP  NUMBER. 


.<:! 


'    *  . 


r '  I 


^<    ! 


t 


,1 


,ij.i 


I  :r 


I,  ..r- 


exercise  in  aritliiiietic  to  a  kindergarten  song  or  a  romp 
in  the  playground. 

Arrested  Development. — Since,  then,  the  natural  ac- 
tion of  tlie  child's  mind  in  gaining  his  first  ideas  of 
niunber  is  attended  with  interest,  it  seems  clear  that 
Avhen  Tinder  the  formal  teaching  of  number  that  inter- 
est, instead  of  being  quickened  and  strengthened,  actu- 
ally dies  out,  the  method  of  teaching  must  be  seriously 
at  fault.  The  method  must  lack  the  essentials  of  true 
method.  It  does  not  stimulate  and  co-operate  witli 
the  rhythmic  movement  of  the  mind,  but  rather  im- 
pedes  and  probably  distorts  it.  The  natural  instinct  of 
number,  which  is  present  in  every  one,  is  not  guided 
by  proper  methods  till  effective  development  is  reached. 
The  native  aptitude  for  number  is  continually  baffled, 
and  an  artificial  activity,  opposed  to  all  rational  devel- 
opment of  numerical  ideas,  is  forced  upon  the  mind. 
From  this  irrational  process  an  arrested  development  of 
the  number  function  ensues.  An  actual  distaste  for  num- 
ber is  created  ;  the  child  is  adjudged  to  have  no  interest 
in  number  and  no  taste  for  mathematics ;  and  to  nature 
is  ascribed  an  incapacity  which  is  solely  due  to  irrational 
instruction.  It  is  perhaps  not  too  much  to  say  that 
nine  tenths  of  those  who  dislike  arithmetic,  or  who  at 
least  feel  that  they  have  no  a'^titude  for  mathematics, 
owe  this  misfortune  to  wrong  teaching  at  first ;  to  a 
method  which,  instead  of  working  in  harmony  with  the 
number  instinct  and  so  making  every  stage  of  develop- 
ment a  preparation  for  the  next,  actually  thwarts  the 
natural  mov^ement  of  the  mind,  and  substitutes  for  its 
spontaneous  and  free  activity  a  forced  and  mechanical 
action  accompanied  with  no  vital    interest,  and   lead- 


'. 


•( 


i-1 


ON   PRIMARY   NUMBER  TEACHING. 


U7 


ing  neither  to  acquired  knowledge  nor  developed 
power. 

Characteristics  of  this  defective  method  have  heen 
frequently  pointed  out  in  the  preceding  pages,  and  it  is 
unnecessary  to  notice  them  here  further  than  to  caution 
the  teacher  against  a  few  of  them,  which  it  is  especially 
necessary  to  avoid. 

Avoid  what  has  been  called  the  "fixed-unit"  meth- 
od. No  greater  mistake  can  be  made  than  to  begin 
wdth  a  single  thing  and  to  proceed  by  aggregating  such 
independent  wholes.  The  method  works  by  fixed  and 
isolated  unities  towards  an  undefined  limit ;  that  is,  it 
attempts  to  develop  accurate  ideas  of  quantity  without 
the  presence  of  that  which  is  the  essence  of  quantity — 
namely,  the  idea  of  limit.  It  does  not  promote,  but 
actually  warps,  the  natural  action  of  the  mind  in  its  con- 
struction of  number  ;  it  leaves  the  fundamental  numer- 
ical operations  meaningless,  and  fractions  a  frowaiing 
hill  of  difliculty.  No  amount  of  questioning  upon  one 
thing  in  the  vain  attempt  to  develop  the  idea  of  "  one," 
no  amount  of  drill  on  two  such  things  or  three  such 
things,  no  amount  of  artificial  analysis  on  the  numbers 
from  one  to  five,  can  make  good  the  ineradicable  defects 
of  a  beginning  which  actually  obstructs  the  primary  men- 
tal functions,  and  all  but  stifles  the  number  instinct. 

Avoid,  then,  excessive  analysis,  the  necessary  conse- 
quence of  this  "rigid  unit"  method.  This  analysis, 
making  appeals  to  an  undeveloped  power  of  numerical 
abstraction,  becomes  as  dull  and  mechanical  and  quite 
as  mischievous  in  its  effects  as  the  "fissure  svstem," 
which  is  considered  but  little  better  than  a  mere  jug- 
glery with  number  symbols. 


n  ,   ■!'. 


mm 


m 


■I  . 


148 


THE  PSYCHOLOGY  OF  NUMBER. 


■ 

^ri. 

A 

'  1 1 

1 

h: 

'1 

f,J 

f' 

H 

.•1 

'    1     ! 

i. 

;, 

ri. 

in 


|Uii 


W'U 


iiiii 


"1 

Hi 

1 

;/  ^^ 

■ 

i  i 

IL 

.1 

Avoid  the  error  of  assuming  tliat  there  are  exact 
numerical  ideas  in  the  mind  as  the  result  of  a  number 
of  things  before  the  senses.  Tliis  ignores  the  fact  tliat 
number  is  not  a  thing,  not  a  property  nor  a  perception 
of  things,  but  the  result  of  the  mind's  action  in  dealing 
with  quantity.  Avoid  treating  numbers  as  a  series  of 
separate  and  independent  entities,  each  of  which  is  to 
be  thoroughly  mastered  before  the  next  is  taken  up. 
Too  much  thoroughness  in  primary  number  work  is  as 
harmful  as  too  little  thorouglmess  in  advanced  work. 

Avoid  on  the  one  hand  the  simultaneous  teaching 
of  tlie  fundamental  operations,  and  on  the  other  hand 
the  teaching  which  fails  to  recognise  their  logical  and 
psychological  connection. 

Avoid  the  error  which  makes  the  "how  many" 
alone  constitute  number,  and  leaves  out  of  account  the 
other  co-ordinate  factor,  "  how  much."  The  nieasuping 
idea  must  always  be  prominent  in  developing  number 
and  numerical  operations.  Without  this  idea  of  meas- 
urement no  clear  conception  of  number  can  be  devel- 
oped, and  the  real  meaning  of  the  various  operations  as 
simply  phases  in  the  development  of  the  measuring 
idea  will  never  be  grasped. 

Avoid  the  fallacy  of  assuming  that  the  child,  to 
know  a  nwnher,  nmst  be  able  to  picture  all  the  num- 
bered units  that  make  up  a  given  quantity. 

Avoid  the  interest-killing  monotony  of  the  Grube 
grind  on  the  three  hundred  and  odd  combinations  of 
half  a  dozen  numbers,  which  thus  substitutes  sheer  me- 
chanical action  for  the  spontaneous  activity  that  simul- 
taneously develops  numerical  ideas  and  the  power  to 
retain  them. 


I 


ON  PRIMARY  NUMBER  TEACHING. 


U9 


national  Method. — The  defects  which  have  been 
enuiTicrated  as  marking  the  "fixed-unit"  method  sug- 
gest the  chief  features  of  the  psycliological  or  rational 
method.  This  method  pursues  a  diametrically  opposite 
course.  It  does  not  introduce  one  object,  theu  another 
"  closely  observed "  object,  and  so  on,  multiplying  in- 
teresting questions  in  the  attempt  to  develop  the  num- 
ber one  from  an  accurate  observation  of  a  single  object. 
It  does  as  Kature  prompts  the  child  to  do  :  it  begins 
with  a  quantity — a  group  of  things  which  may  be  meas- 
ured— and  makes  school  instruction  a  continuation  of 
the  process  by  which  the  child  has  already  acquired 
vague  numerical  ideas.  Under  Nature's  teaching  the 
child  does  not  attempt  to  develop  the  number  one  by 
close  observation  of  a  single  thing,  for  this  observation, 
however  close,  will  not  yield  the  number  one.  He  de- 
velops the  idea  of  one,  and  all  other  numerical  ideas, 
through  the  measuring  activity  ;  he  counts,  and  thus 
measures,  apples,  oranges,  bananas,  marbles,  and  any 
other  things  in  which  he  feels  some  interest.  ^NTature 
does  not  set  him  upon  an  impossible  task — i.  e.,  the 
getting  of  an  idea  under  conditions  which  preclude  its 
ac(piisition.  She  does  not  demand  numerical  abstrac- 
tion and  generalization  wlien  there  is  nothing  before 
him  for  this  activity  to  work  upon.  Let  the  actual 
work  of  the  schoolroom,  tlierefore,  be  consistent  with 
tlie  method  under  which  by  Nature's  teaching  the  child 
has  already  secured  some  development  of  the  number 
activity. 

In  all  psychical  activity  every  stage  in  the  develop- 
ment of  an  instinct  prepares  the  way  for  the  next  stage. 
The  child's  number  instinct  bei^ins  to  show  itself  in  its 


I 


(i  i 


m 


f 


ifc 


v^f 


\ 
i 


m\ 


in 

J  ■*       I 


It  ;■     1 


•    M 


150 


THE   PSYCHOLOGY  OF  NUMBER. 


working  upon  continuous  quantity — that  is,  a  whole 
requiring  measurement.  Every  successive  step  in  the 
entire  course  of  development  should  harmonize  with 
this  initial  stage.  To  get  exact  ideas  of  quantity  the 
mind  must  follow  Nature's  estahlished  law;  must  meas- 
ure quantity ;  must  break  it  into  parts  and  unify  the 
parts,  till  it  recognises  the  one  as  many  and  the  many 
as  one.  There  can  be  no  possible  numerical  abstraction 
and  generalization  without  a  quantity  to  be  measured. 
AVhere,  then,  does  the  "  single  closely  observed  object " 
come  in  as  material  for  this  parting  and  wholing  ? 

Beginning  with  a  group  is  in  harmony  with  Nature's 
method  ;  promotes  the  normal  action  of  the  mind  ;  gives 
the  craving  numerical  instinct  something  to  work  upon, 
and  wisely  guides  it  to  its  richest  development.  This 
psychological  method  promotes  the  natural  exercise  of 
mental  function  ;  leads  gradually  but  with  ease  and  cer- 
tainty to  true  ideas  of  number ;  secures  recognition  of 
the  unity  of  the  arithmetical  operations  ;  gives  clear 
conceptions  of  the  nature  of  these  operations  as  succes- 
sive steps  in  the  process  of  measurement ;  minimizes  the 
difficulty  with  which  multiplication  and  division  have 
hitherto  been  attended  ;  and  helps  the  child  to  recognise 
in  the  dreaded  terra  incognita  of  fractions  a  pleasant 
and  familiar  land. 

Forming  the  Habit  of  Parting  and  Wholing. — 
The  teacher  should  from  the  first  keep  in  view  the  im- 
portance of  forming  the  hahit  of  parting  and  wholing. 
This  is  the  fundamental  psychical  activity ;  its  goal  is  to 
grasp  clearly  and  definitely  by  one  act  of  mind  a  whole 
of  many  and  defined  parts.  This  primary  activity  work- 
ing upon  quantity  in  the  process  of  measurement  gives 


ON   PRIMARY  NUMBER  TEACHING. 


151 


rise  to  numerical  relations  ;  the  incoherent  whole  is 
made  definite  and  unified — becomes  the  conception  of  a 
unity  composed  of  units.  Every  right  exercise  of  this 
activity  gives  new  knowledge  and  an  increase  of  analytic 
power.  At  last  the  hahit  of  numerical  analysis  is 
formed,  and  when  it  is  found  requisite  to  deal  with 
quantity  and  quantitative  relations,  the  mind  always 
conceives  of  quantity  as  made  up  of  parts — measuring 
units;  not  invariable  units,  but  units  chosen  at  pleasure 
or  convenience  ;  parts,  given  by  the  necessary  activity 
of  analysis,  a  whole  from  the  parts  by  the  necessary  ac- 
tivity of  synthesis.  This  means  that  always  and  inevi- 
tably from  first  to  last  the  process  of  fractioning  is 
present. 

A  Consti'uctwG  Process. — This  wholing  and  parting, 
as  far  as  possible,  should  be  a  constructive  act.  The 
physical  acts  of  separating  a  whole  into  parts  and  re- 
uniting the  parts  into  a  whole  lead  gradually  to  the 
corresponding  mental  process  of  number  :  division  of  a 
whole  into  exact  parts,  and  the  reconstruction  of  the 
parts  to  form  a  whole.  It  can  not  be  said  that  even  the 
physical  acts  are  wholly  mindless,  for  even  in  these 
acts  there  must  be  at  least  a  vague  mental  awareness  of 
the  relation  of  the  ])arts  to  one  another  and  to  the  whole. 
These  physical  acts  of  wholing  and  parting  under  wise 
direction  lead  quickly,  and  with  the  least  expenditure 
of  energy,  to  clear  and  definite  percepts  of  related 
things,  and  finally  to  definite  conceptions  of  number. 
The  child  should  be  required  to  exercise  his  activity,  ^o 
do  as  much  as  possible  in  the  process,  and  to  notice  and 
state  what  he  is  really  doing.  lie  should  actually  apply, 
for  instance,  the  measuring  unit  to  the  measured  quan- 


I  \\ 


tm 


m 

;  "  . !  f ) 

it-';-' 


iJK 


'  t  , 


0 

I'*  i' 


,;( 


^i 


ii,  1 


li'- 


;  i'l 


Hlr* 


^5? 


152 


THE  PSYCHOLOGY  OF  NUMBER. 


''I  A 


.^1  \ 


■  ■ 

f 

m 
-J 


w 


tity.  If  the  foot  is  measured  by  two  G-incli  or  three  4- 
inch  or  four  3-iiicli  units,  lot  liiiii  lirst  apply  the  miniber 
of  actual  units — two  G-inch,  three  4-ineh,  and  four  3-incli 
units — to  make  up  the  foot,  and  so  on.  By  using  the 
actual  number  of  parts  required  he  will  have  a  more 
definite  idea  of  the  construction  of  a  whole  than  if  he 
simply  applies  one  of  the  measuring  units  the  necessary 
number  of  times.  This  operation  with  the  actual  units 
should  precede  the  operation  by  which  the  whole  is 
mentally  constructed  by  applying  or  repeating  the  sin- 
gle unit  of  measurement  the  required  number  of  times. 
It  is  the  more  concrete  process,  and  is  an  effective  exer- 
cise for  the  gradual  growth  of  the  more  abstract  times 
or  ratio  idea. 

When  the  child  actually  uses  the  1-incli  or  the  3-inch 
unit  to  measure  the  foot,  his  ideas  of  these  units  as  well 
as  of  the  measured  whole  are  enlarged  and  defined.  He 
applies  the  inch  to  measure  the  foot,  and  this  to  meas- 
ure the  yard,  and  the  yard  to  measure  the  length  of  the 
room  and  other  quantities.  Let  him  freely  practise  this 
constructive  activity,  thus  practically  applying  the  psy- 
chological law,  "  Know  by  doing,  and  do  by  knowing." 
The  2-inch  square  is  separated  into  four  inch  squares,  or 
sixteen  half-inch  squares,  and  these  measuring  units  are 
put  together  again  to  form  the  whole.  Similarly  a  rec- 
tangle 2  inches  by  3  inches,  for  example,  is  divided  into 
its  constituent  inch  squares  or  half-inch  squares,  and 
again  reconstructed  from  the  ])arts.  A  square  is  divided 
into  four  right-angled  isosceles  triangles,  into  eight 
smaller  triangles,  and  the  parts  rhythmically  put  to- 
gether again. 

Value  of  Kinde^'garten  Constructions. — In  this  con- 


ON   PltlMARY   NUMBER  TEACHING. 


153 


nection  it  may  be  noted  that  most  of  the  exercises  of 
the  kindergarten  can  be  etl'ectively  used  for  training 
in  number.  The  constrnctive  exercises  which  are  so 
])rominent  a  featnre  in  tlie  kindergarten  are  admira- 
bly adapted  to  lead  gradually  to  mathematical  abstrac- 
tion and  generalization,  ^o  doubt  much  has  been  done 
in  this  direction,  but  much  more  could  be  done  were 
the  teacher  versed  in  the  psychological  method  of  deal- 
ing with  number.  No  one  questions  the  general  value 
of  kindergarten  i  '-.lining,  which  on  the  whole  is  founded 
on  sound  psycliological  principles  ;  but,  on  the  other 
hand,  no  educati  >nal  psychologist  doubts  that  its  phi- 
losophy as  commonly  understood  needs  revision,  and 
that  its  methods  are  capable  of  improvement.  If  its 
aim  is,  as  it  should  be,  an  effective  preparation  of  the 
child  for  his  subsequent  educational  course,  it  is  thought 
that  its  practical  results  are  far  from  what  they  ought 
to  be.  It  is  ofti  n  maintained  with  considerable  force 
that  kinder'^arte  I  methods  should  be  introduced  into 
the  primary  and  even  higher  schools.  On  the  other 
\\pj\L  ooiiie'hing  might  be  said  with  a  good  show  of 
reasoii  ia  favour  of  introducing  primary  and  grammar 
school  methods  into  the  kindergarten.  What  is  radi- 
cally sound  in  the  kindergarten  methods  will  harmonize 
with  what  is  radically  sound  in  the  methods  of  the  pub- 
lic school.  On  the  other  hand,  what  is  psychologically 
sound  in  the  methods  of  the  public  school  should  at 
least  influence  the  aims  and  methods  of  the  kindei'gar- 
ten.  Is  the  p'^esent  kindergarten  training,  speaking 
generally,  really  the  best  preparation  for  the  training 
given  in  a  thoroughly  good  public  school  ?  The  func-  . 
tion  of  such  a  school  is  to  give  the  best  possible  prepa- 


^ 

5^1 

:|! 

I  \ 

M 


\  M\ 


■.I  ■ . 


.•y 


i" 

M 

'i' 

h 

fi 

1, 

;  i  ' 

1 , . 

1 

I'.'l    :■ 


ri'ii 


li 


154 


THE   PSYCHOLOGY  OP  NUMBER. 


ration  for  life  by  means  of  studies  and  discipline,  wliich, 
as  far  as  inevitable  limitations  permit,  seciiro  at  the 
same  time  the  best  possible  development  of  character. 
Amon£«:  these  studies  the  three  IV s  must  always  hold 
a  prominent  place,  in  spite  of  theories  which  seem  to 
assume  that  language,  the  complement  of  man's  reason, 
and  number,  the  instrument  of  man's  interpretation  and 
mastery  of  the  physical  world,  are  not  essential  to  hu- 
man advancement,  and  may  therefore  be  degraded  from 
the  central  position  which  they  have  long  occupied  to 
one  in  which  they  are  the  subjects  of  merely  haphazard 
and  disconnected  teaching. 

The  practical  methods  founded  on  these  theories 
seem  to  treat  the  world  of  Nature  as  one  whole,  which 
even  the  child  may  grasp  in  its  infinite  diversity  and 
total  unity.  The  "  flower  in  the  crannied  wall "  is 
made  the  central  point  around  which  all  that  is  know- 
able  is  to  be  collected.  But  as  the  human  mind  is  lim- 
ited, and  must  move  obedient  to  the  law  of  its  consti- 
tution, the  theories  and  metliods  wdiich  overlook  these 
facts  are  not  likely  permanently  to  prevail ;  and  the  old 
subjects  that  have  stood  the  test  of  time  will  no  doul)t 
stand  the  test  of  the  most  searching  psychological  inves- 
tigation, and  regain  their  full  recognition  as  the  "core" 
subjects  of  the  school  curriculum. 

Does  the  kindergarten,  then,  accomplish  all  tliat  may 
be  done  as  a  preparation  for  such  a  curriculum  ?  It  is 
to  be  feared  that  with  regard  to  many  of  them  the  an- 
swer must  be  in  the  negative  ;  and  this  is  perhaps  es- 
pecially true  concerning  the  subject  of  arithmetic.  We 
have  known  the  seven-year-old  "  head  boy "  of  a  kin- 
dergarten, conducted  by  a  rioted  kindergarten  teacher, 


ON   I'RLMARY  NUMBER  TExVCIIING. 


155 


m 


who  could  not  recognise  a  (jiiaiitity  of  tliroe  tliinj:;s  with- 
out countiiiii^  thorn  hv  ones.  Jjuin<«^  asked  to  heirin  tlic 
construction  of  a  certain  form  at  a  distance  of  three 
inches  from  tlie  edge  of  the  table,  he  invariably  had 
to  count  off  carefully  the  inch  spaces — a  clear  proof,  it 
is  thought,  that  his  training  had  not  l)een  the  best  pos- 
sible preparation  for  the  arithmetical  instruction  of  the 
higher  schools. 

It  is  certain  that  arithmetical  instruction  in  the 
higher  grade  of  school  may  be  greatly  improved  ;  it  is 
alike  certain  that  as  a  preparation  for  this  better  in- 
struction the  ti'aining  of  the  kindei-garten  also  may  be 
greatly  improved  ;  and  there  is  every  reason  to  believe 
that  with  this  improvement  its  rational  training  in 
other  things,  its  ethical  aim,  its  educative  interest,  and 
its  character-forming  spirit,  would  be  materially  en- 
hanced. In  some  kindergartens,  at  least,  the  monotony 
of  continuous  play  would  be  pleasurably  relieved  by  a 
little  recreation  at  work.  It  is  certain  that  the  interest 
associated  with  many  exercises  necessarily  connected 
with  the  number  activity,  especially  the  constructive 
and  analytic  processes  of  the  kindergarten,  can  be  made 
under  right  teaching  the  menus  by  which  numerical 
ideas  may  be  gradually  and  pleasantly  worked  out. 
The  little  builder  of  many  forms  of  beauty  and  utility 
would  in  due  time  find,  when  the  inevitable  and  harder 
tasks  began,  that  he  had  been  building  better  than  he 
knew.  There  is  surely  something  lacking  either  in  the 
kindergarten  as  a  preparation  for  the  primary  school, 
or  in  the  primary  school  as  a  continuation  of  the  kin- 
dergarten, when  a  child  after  full  training  in  the  kinder- 
garten, together  with  two  years'  work  in  the  primary 


ll'ili 


I 


150 


THE  PSYCriOLOGY  OF   NUMBER. 


1^1 


WW 


school,  is  considered  able  to  undertake  nothing  beyond 
the  "number  20."  It  might  reasonably  be  maintained 
that,  under  rational  and  therefore  pleasurable  training 
of  the  number  instinct  in  the  kindergarten,  the  child 
ou<>;ht  to  be  aiithmeticallv  strong  enouii-li  to  make  ini- 
mediate  accjuaintance  with  the  number  20,  and  rapidly 
accjuire — if  he  has  not  already  ac(|uired — a  working  con- 
ce])tioii  of  much  larger  numbers. 

Iin2)ortant  Points  of  Bational  Method. — In  apply- 
ing!: the  rational  method  of  teaching  arithmetic  there  are 
important  things  that  the  teacher  must  keep  in  view 
if  he  is  to  phI  the  child's  mind  to  v;ork  fioely  and  nat- 
urally in  tit;  evolution  of  mmiber.  The  child's  niind 
must  be  guided  aloui:^  the  lines  of  least  resistance  to 
the  true  idea  of  n%iiiJjer.  This  movement,  in  the  very 
nature  of  things,  must  be  slow  as  compared  with  the 
gathering  of  sense  facts  ;  but  under  the  psychological 
method  it  may  be  sure  and  pleasurable.  The  result 
aimed  at  can  not  be  reached  b^'  banishinj>:  the  word 
times  from  arithmetic  ;  nor  [)y  working  continually 
with  indefinite  units  of  measure  ;  nor  by  exclusive  at- 
tention to  manual  occu])ations  under  the  vague  idea 
that  physical  separation  of  things  is  analysis  of  thought ; 
nor  by  making  counting — emphasizing  the  vague  how 
many — the  single  purpose,  and  unmeasured  units  the 
sole  matter  of  the  exercises,  to  the  exclusion  uf  the  how 
7nvch,  and  the  measuring  idea  which  is  the  essence  of 
number ;  nor  by  substituting  for  the  rhythmic  and 
bpontaneous  action  of  the  child's  mind  in  dealing  with 
wholes,  both  qualitative  and  (juantitative,  a  minute  and 
formal  analysis  which  properly  finds  ])lace  only  in  a 
riper  stage  of  mental  growth  ;  nor  by  any  amount  of 


i 


ON   PRIMARY   NUMBER  TEACIIINO. 


15' 


drill,  liowever  industrious  and  deviceful  the  drill-mas- 
ter, which  substitutes  mechanical  action  and  factitious 
interest  for  spontaneous  action  and  intrinsic  interest, 
the  very  life  of  the  self-developing  soul. 

Kumber  is  the  measurement  of  quantity,  and  there- 
fore the  only  solid  basis  of  method  and  sure  guide  for 
the  teacher  is  the  measurvmj  idea. 

1.  The  Memured  Whole. — The  factors  in  number 
are,  as  before  shown,  the  unity  (the  whole  of  quantity) 
to  be  measured,  the  unit  of  measurement,  and  t^e  times 
of  its  repetition — the  number  in  the  strictly  mathemat- 
ical and  psychological  sense  of  the  word.  The  teacher 
must  bear  in  mind  the  distinction  between  unity  and 
unit  as  fundamental.  The  entire  dilference  between 
a  good  method  and  a  bad  method  lies  here,  because  the 
essential  principle  of  mnnber  lies  here.  Vague  unity, 
units,  defined  unity,  is  the  secjuence  as  determined  by 
psychological  law.  In  the  child's  lirst  deahng  with 
number  there  must  be  the  grouj)  of  th:  2:s,  tin  whole 
of  quantity  to  start  from  ;  and  in  every  step  of  the  ini- 
tial stage  the  idea  of  a  whole  to  be  measured  is  to  be 
kept  prominent.  In  addition,  there  is  a  whole  (the 
sum)  to  be  made  more  definite  by  putting  together  its 
component  parts  (addends) — not  equal  measKrlna  units, 
but  each  part  defined  l)y  a  common  unit — so  as  to  com- 
ph^tcly  define  the  quantity  in  terms  of  this  specifically 
defined  unit.  In  subtraction  we  have  a  given  quantity 
(minuend)  and  a  component  part  (subtrahend)  of  it  to 
find  the  other  conq)oneMt  (remainder) — a  process  which 
lielps  to  a  more  definite  idea  of  the  given  whole,  and 
especially  makes  explicit  the  vague  idea  of  the  "remain- 
der" with  ;vhich  we  began.     In  nmlti])lication  there  is 


M    m 


,:'■■'  5; 


I 


r-M 


158 


THE  PSYCHOLOGY  OP  NUMBER. 


1 
1 

^ 

•  1 

'.  ■  1 

1 

:f      1 

» i 


given  a  quantity  (mnltiplicand)  defined  by  a  measuring 
unit  and  the  times  (multiplier)  of  its  repetition  ;  and 
the  process  makes  the  quantity  articulately  defined  (in 
the  product)  by  substituting  a  more  familiar  unit  for 
the  derived  unit  of  measurement ;  in  other  words,  by 
expressing  the  quantity  in  terms  of  the  j^rimary  unit 
by  which  the  derived  unit  itself  is  measured.  In  divi- 
sion we  have  a  whole  quantity  given  (dividend),  and 
one  of  two  related  measuring  parts  (the  divisor)  to 
find  the  other  part  (quotient),  and  the  operation  makes 
clearer  the  whole  magnitude,  and  at  the  same  time  makes 
the  first  vague  idea  of  the  other  measuring  part  (quo- 
tient) perfectly  definite.  Briefly,  in  all  numerical  oper- 
ations there  is  some  magnitude  to  be  definitely  deter- 
mined in  numerical  terms,  and  the  arithmetical  opera- 
tions are  simply  related  steps  expressing  tlie  correspond- 
ing stages  of  the  mental  movement  by  which  the  vague 
whole  is  made  definite.  Keep  clearly  in  mind,  there- 
fore, the  inclusive  magnitude  from  which  and  within 
which  the  mental  movement  takes  place — which  justi- 
fies and  gives  meaning  to  both  the  psychical  process  and 
the  arithmetical  operation. 

2.  The  U?nt  of  Measure — Its  True  Functw\ — 
From  the  vague  unity,  through  the  vnll.^,  to  the  deii- 
nite  unity,  the  snm^  is  the  law  of  mental  movement. 
The  second  point  of  essential  importance  is  to  make 
clear  the  idea  of  the  unit  of  measure.  More  than  half 
the  difiiculty  of  the  teacher  in  teaching,  and  the  learner 
in  learning,  is  due  to  misconce])tion  of  what  the  ''unit'' 
really  is.  It  is  not  a  single  unmeasured  object ;  it  is 
not  even  a  single  defined  or  measured  thing  ;  it  is  any 
measuring  jjart  by  which  a  quantity  is  numerically  de- 


:!l 


ON  PRIMARY  NUMBER  TEACHING. 


159 


fined ;  it  is  (in  the  crude  stage  of  measurement)  one  of 
the  like  things  used  to  measure  a  collection  of  the 
things ;  it  is  (in  the  second  or  exact  stage  of  measure- 
ment) one  of  the  equal  parts  used  to  measure  an  ex- 
actly measured  quantity.  It  is  one  of  a  necessarily 
related  many  constituting  a  whole. 

It  is,  therefore,  an  utterly  false  method  to  begin  with 
an  isolated  object — false  to  the  fact  of  measurement, 
false  to  the  free  activity  of  the  mind  in  the  measurin<r 

</  CD 

process.  IS'or  is  the  defect  to  be  remedied  by  intro- 
ducing another  isolated  object,  then  another,  and  so  on. 
The  idea  of  a  unit  can  begin  only  from  analysis  of  a 
whole ;  it  is  completed  only  by  relating  the  part  to  the 
whole,  so  that  it  is  linally  conceived  at  once  in  its  isola- 
tion and  in  its  unity  in  the  whole.  Not  only  do  we  not 
begin  with  a  single  object  and  "  develop  one,"  but  also 
even  in  beginning,  as  psychology  demands,  with  a  group 
of  objects,  we  are  not  to  begin  with  the  single  object  to 
measure  the  quantity — at  least  we  are  not  to  emphasize 
the  single  object  as  pre-eminently  the  measuring  unit. 
"We  separate  twelve  beans,  for  (example,  not  into  12  parts, 
but  into  2  parts,  then  3  parts,  etc. ;  that  is,  w^e  measure 
by  6  beans,  by  4  beans,  by  3  beans,  by  2  beans  ;  and  the 
resulting  mimhers  for  the  one  measured  ^juantity  ai-e 
two,  three,  four,  six  ;  and  each  of  the  measuring  parts 
'o  wliicli  the  numbers  are  ap])1ie(l  is  a  unit^  is  one.  So 
in  building  up  the  measured  foot  with  G-inch,  4-inch, 
3-inch,  2-inch  measures — each  in  turn  measuved  off  in 
inches — there  are  two  onen^  three  ones^  four  ones,  six 
ones.  The  point  to  be  kept  in  view  is  to  prevent  the 
mischievous  error  of  regarding  the  unit  as  a  single  uh- 

JECT,  a  lixed  qualitative  or  an  indivisible  quantitative 
12 


!i    ' 


:| 


!• 


I 


i-i 


160 


THE  PSYCHOLOGY  OF  NUMBER. 


.1   '' 


I*! 


unity,  instead  of  simply  a  measuring  part — a  means  of 
measnrinc;  a  ma2:nitude. 

The  Unit  itself  Meamired. — As  necessary  to  the 
growth  of  the  true  conception  of  unit  as  a  measuring 
part,  the  idea  of  the  unit  as  a  unity  of  measured  parts 
must  be  clearly  brought  out.  The  given  quantity  is 
measured  by  a  certain  unit ;  this  unit  itself  is  a  quantity, 
and  so  is  made  up  of  measuring  parts.  This  idea  must 
be  used  from  the  beginning ;  it  is  absolutely  essential  to 
the  clear  idea  of  the  unit,  and  of  number  as  measure- 
ment of  quantity.  Beginning  with  a  group  of  12  ob- 
jects requiring  measurement,  or  with  coanters  repre- 
senting such  objects,  we  have  them  counted  oft'  into 
two  equal  parts,  noting  the  relation  of  the  parts  to  one 
another  and  to  the  whole  ;  then  each  of  these  two  units 
(half  of  the  given  whole)  is  counted  oft  into  two  equal 
parts,  and  the  relation  of  these  minor  parts  to  each 
other  and  to  the  whole  they  compose  is  noticed  ;  then 
each  of  the  first  units  of  measurement  (halves  of  the 
given  whole)  is  counted  off  into  three  equal  parts,  and 
their  relation  to  one  another  and  to  the  whole  which 
they  make  is  carefully  observed  ;  and  so  on,  with  similar 
exercises  in  parting  and  wholing.  Such  constructive 
exercises  help  in  the  growth  of  the  true  idea  of  the 
unit  as  a  measuring  part,  which  is  or  may  be  itself 
measured  by  other  units.  But  the  true  idea  of  the  es- 
sential property  of  the  unit — its  measuring  function — 
can  be  fully  developed  only  by  exercises  belonging  to 
the  second  stage  of  measurement,  in  which  exact  and 
equal  units  a^e  used  for  precise  measurement.  These 
measurements  of  groups  of  like  things  (apples,  oranges, 
etc.)  by  groups  which  are  themselves  measured  by  still 


i'f 


ON  PRIMARY  NUMBER  TEACHING. 


161 


smaller  groups,  mmst  be  supplemented  by  the  use  of  ex- 
actly measured  quantities — quantities  defined  by  eqnal 
units,  wliicli  in  turn  are  measured  by  other  equal  units. 
Without  such  exercises  there  can  be  no  adequate  con- 
ception of  measurement,  or  of  number  as  the  tool  of 
measurement,  or  of  the  real  meaning  of  multiplication 
and  division,  and  especially  of  fractions,  the  full  and 
precise  statement  of  the  measuring  process.  To  free 
arithmetic  from  the  tyranny  of  irrational  method,  an 
indispensable  step  is  the  emancipation  of  the  unit  from 
the  cast-iron  fetters  which  have  paralyzed  its  weasur- 
ing  function. 

3.  The  Idea  of  Times. — With  the  intelligent  use  of 
these  constructive  exercises  to  make  clear  the  idea  of 
tlie  unit,  there  is  necessarily  growth  towards  recog- 
nition of  the  times  of  repetition  of  the  unit  to  make 
up  some  magnitude — towards,  that  is,  the  true  idea  of 
number.  To  discuss  the  evolution  of  this  idea  would 
be  to  repeat  in  the  main  what  has  been  laid  down  in 
the  preceding  paragraphs.  It  is  therefore  necessary 
only  to  state  explicitly  the  chief  things  to  be  considered 
as  bearing  upon  the  natural  growth  of  the  idea  of  times 
— i.  e.,  of  nnmher  in  the  strict  sense  of  the  word. 

{a)  The  preliminary  operations  as  already  illustrated 
— dealing  with  groups  of  like  things,  and  so  leading  to 
a  working  idea  of  the  unit  as  meamwinfj  part — are  to  be 
supplemented  by  constructive  acts  with  exactly  meas- 
ured quantities.  The  measured  whole  must  be  analyzed 
into  its  measured  units,  and  again  built  up  from  these 
parts.  For  example,  exercises  such  as  tlie  quantity  12 
apples  measured  by  the  unit  4  apples,  by  the  unit  3 
apples,  etc.,  must  be  supplemented  by  exercises  such 


! 

> 


§• 


i '  1 


f '  : 


I  f 


»     t 


.(  : 


I' « 


■1! 


hi; 


I, ,  •  I 


>l    t 


rill' 


162 


THE  PSYCHOLOGY  OF  NUMBER. 


as  the  quantity  12  inches  measured  by  the  unit  4 
inches,  by  the  unit  3  inches,  etc. ;  or  the  quantity  20 
cents  measured  by  the  unit  10  cents,  by  the  unit  5 
cents,  etc.  The  movement  towards  the  real  number 
;idea  began  in  operations  with  undefined  units,  and  is 
strengthened  by  these  supplementary  exercises  with  ox- 
Jactly  measured  quantity  ;  thei-e  is  a  more  rapid  growth 
towards  the  numerical  discriminating  and  unifying 
power,  {h)  Count  by  o}ic%  but  not  necessarily  by  sin- 
gle things  ;  in  fact,  to  avoid  the  fixed  unit  error,  do 
not  begin  with  counting  single  things.  The  12  things  in 
the  group  have  been  measured  off,  for  example,  into  four 
groups,  or  into  three  groups  ;  these  are  units,  are  ones, 
and  in  counting  there  is  a  first  one,  a  second  oi^e,  a  third 
07ie — that  is,  in  all  ''three  times^'  one;  and  so  with  the 
four  ones  when  the  quantity  is  divided  into  four  equal 
parts.  Proceed  similarly  with  exactly  measured  quanti- 
ties :  the  four  f3-inch  ones  or  the  six  2-incli  ones  making 
up  the  linoai  .  )ot,  or  other  exactly  measured  quantity. 
As  before  said,  the  child  first  of  all  sees  related  things, 
and  with  the  repetition  of  the  exercises — parting  and 
wholing — begins  to  feel  the  relations  of  things,  and  in 
due  time  consciou>lv  recoiniises  these  relations,  and  the 
goal  is  at  last  reached — a  definite  idea  of  number. 

{c)  Use  the  ActiKilTn  (ti<. — In  these  constructive  proc- 
esses let  the  child  at  first  use — as  before  suggested — the 
actual  concrete  units  to  miike  up  or  etpial  the  measured 
([uantity  ;  then  apply  the  shxjle  r'oncrete  unit  the  requi- 
site number  of  "  times."  It  first  case,  in  measuring, 
for  example,  a  length  of  l:i  feet,  four  actual  units  of 
measure  (3  feet)  are  put  toi/'^'^'er  to  equal  the  12  feet; 
in  the  second  case,  one  unit  ^  applied,  laid  down,  and 


1 


I 


ON   PIlIMArtY  NUMBER  TEACHING. 


103 


taken  up  three  times.  This  application  of  tlie  single 
unit  so  many  times  is  an  important  step  in  the  process 
of  numerical  abstraction  and  generalization  ;  it  is  from 
.the  less  abstract  and  more  concrete  to  the  more  abstract 
and  less  concrete.  It  may  be  noted,  also,  that  the  other 
senses,  especially  the  sense  of  hearing,  may  be  made  to 
co-operate  with  sight  in  the  evolution  of  the  thnes  idea. 
Appeal  by  a  variety  of  examples  to  the  trusty  eye, 
but  appeal  also  to  the  trusty  ear — strokes  on  a  bell, 
taps  on  the  desk,  uttered  syllables,  etc.  Here,  as  in  all 
other  cases,  we  do  not  confine  ourselves  to  single  bell- 
strokes  or  svllables  ;  we  count  the  number  of  double 
strokes,  triple  strokes  ;  of  double  and  triple  syllables, 
as,  for  example,  oh^  oh ^  oli^  oh;  oh,  oh — i.  e.,  3  counts 
of  two  sounds  each,  etc. 

Counting  and  Measuring. — In  the  separating  and 
combining  processes  referred  to,  counting  goes  on. 
This  is  at  first  chiefly  mechanical,  and  care  must  be 
taken  in  the  interest  of  the  number  idea  to  make  it 
become  rational.  Through  practice  in  parting  and 
wholing  the  idea  of  the  function  of  the  unit  is  grad- 
ually formed  ;  it  is  the  concrete,  spatial  thing  used  to 
measure  quantity.  The  point  is,  not  to  neglect  either 
the  sy)atial  element  or  the  other  essential  factor  in  num- 
ber, the  counting,  the  actual  relating  process. 

In  the  method  of  number  teachint!:  usuallv  followed, 
counting  is  the  prominent  thing,  to  the  almost  total 
exclusion  of  the  measuring  idea  ;  the  emphasis  is  upon 
the  how  manv,  with  but  little  attention  to  the  how  much. 
But  the  counting  is  largely  mechanical.  There  is  a  repe- 
tition of  names  without  definite  meaning.  The  child  is 
groping  his  way  toward    the  light.     He  can  not  help 


I'll 


■■pp 


w 


M 


I    r 

.1:      M 


l<  '    I 


I'    I 


164 


THE  PSYCHOLOGY  OF  NUMBER. 


feeling,  as  lie  counts  liis  units,  tliat  one,  two,  three  is 
not  so  much — because  it  is  not  so  many — as  one,  two, 
three,  four.  These  first  vague  ideas  must  he  made 
clear  and  definite  ;  the  natural  movement  of  the  mind 
is  aided  by  the  proper  presentation  of  right  material ; 
the  initial  mechanical  operation  of  naming  the  units  in 
order  gives  place  to  an  intelligent  relating  of  the  units 
to  one  another,  and  finally  to  a  conscious  grasp  of  the 
relation  of  each  to  the  unified  whole ;  the  counting — 
one,  two,  three,  etc. — is  now  a  rational  process. 

/So  much;  so  many. — In  tlie  development  of  this 
rational  process  there  must  be  no  divo rr;e  between  the 
how  much  and  the  how  many,  between  the  measuring 
process  and  the  results  of  measurement.  The  so  nmch 
is  determined  only  by  the  so  many,  and  the  so  many 
has  significance  only  from  its  relation  to  the  so  much. 
These  are  co-ordinate  factors  of  the  idea  of  number  as 
measurement.  Now,  the  development  of  counting — 
determining  the  how  many  that  defines  the  how  much 
— is  aided  by  symmetrical  arrangements  of  the  units  of 
measure  (see  page  34).  The  child  at  first  counts  the 
units  one,  two,  .  .  .  six,  with  only  the  faintest  idea  of 
the  relations  of  the  units  in  the  numbers  named.  Both 
the  analytic  and  relating  activities  are  greatly  aided  by 
the  rhythmic  grouping  of  tlie  units  of  measure,  or  of 
the  counters  used  to  represent  them  ;  the  mastery  of 
the  number  relations  (of  both  addition  and  multiplica- 
tion) as  so  many  units  making  up  a  quantity,  becomes 
much  easier  and  more  complete.  Thus,  when  exercises 
in  parting  and  wholing  (accompanied  witli  counting)  a 
quantity,  say  a  length  of  12  inches,  have  given  rise  to 
even  imperfect  ideas  of  unit  of  measurement  and  times 


:ri; 


■■^ 


ON   PULMaRY  number  TEACHING. 


1G5 


of  repetition,  the  symmetric  forms  may  be  used  with 
great  advantage;  indeed,  they  may  be  used  in  the  exer- 
cises from  the  first.  We  lave  counted,  according  to  the 
unit  of  measure  used,  one  oart,  two  parts ;  one  part,  two 
parts,  three  parts,  etc.  Both  the  times  and  the  unit 
values  are  more  easily  grasped  through  the  number  forms ; 
for  example,  six,  one  of  the  two  measuring  units,  may  be 
shown  as  a  whole  of  related  units  (threes,  twos,  ones)  as 

in  the  arrangement,   1,1;   and  so  on  with  the  whole 

quantity  and  all  its  minor  parts  (addition)  and  repeated 
units.  Heal  meaning  is  given  to  the  operation  of  count- 
ing when,  instead  of  using  unarranged  units,  we  have 
the  rhythmic  arrangement : 


e    o 


•  0 

o 

•  e 


o    o 


•    • 


e    o 

•    • 

Six,  five,  four,        three,         two,        one. 

The  actual  values  of  the  measuring  units,  and  the  mean- 
ing of  counting — necessarily  related  processes — are  fully 
brought  out.  Six  is  at  last  perceived  as  six  without  the 
necessity  of  counting. 


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CHAPTER  IX. 

ON    PRIMARY    NUMJ}ER    TEACHING. 

lielation  lettceen  Times  and  Parts. — AYitli  the 
growth  of  tlie  idea  of  the  unit  as  a  measure  itself 
measured  by  minor  units,  and  of  number  as  indicating 
times  of  repetition  of  a  unit  of  measure,  there  is  grad- 
ually developed  a  clear  idea  of  the  relation  between  the 
value  of  the  actual  measuring  unit  (as  made  up  of  minor 
units)  and  the  number  of  them  in  tlie  given  quantity  ;  in 
other  words,  of  the  relation  between  the  number  of  de- 
rived units  in  the  quantity  and  the  number  of  primary 
units  in  the  derived  unit.  A  common  error,  as  has  been 
often  pointed  out,  is  that  of  making  too  broad  a  distinc- 
tion between  these  related  factors  in  the  measuring  proc- 
ess. They  are  said  to  be  totally  different  conceptions. 
It  has  been  shown  that  they  are  absolutely  inseparable. 
They  are,  in  any  and  every  case,  two  aspects  of  the  same 
measurement.  The  direct  unit  in  a  given  measurement 
is  not  wholly  concrete  ;  it  is  a  quantity  measured  by  a 
number  of  other  units  ;  and  so  it  involves,  as  every  meas- 
ured quantity  involves,  the  space  element  in  the  single 
concrete  (minor)  unit,  and  the  abstract  element  in  the 
number  applied  to  the  unit.  AVhen  we  speak  of  the 
"  size"  of  the  numbered  parts  (derived  units)  composing 

a  given  quantity,  we  mean  the  number  of  minor  units 

166 


ON   PRIMARY   NUMBER  TEACHING. 


ir.7 


of  wliicli  one  part  is  composed.  Wlicii,  for  instance, 
we  conceive  of  $15  as  measured  by  the  nnit  J^3,  we  get 
the  number  live  ;  when  we  are  required  to  divide  lj>15 
into  live  equal  parts,  we  are  searching  for  the  "  size  " 


of  the  measuring  unit — i.  e.,  for  the  numerical 
value  of  the  unit  in  terms  of  the  minor  unit  ($1)  by 
wliich  it  is  measured. 

The  relation,  then,  between  "times  and  parts"  is 
the  relation  between  the  number  of  derived  units  in  the 
measured  quantity  and  the  number  of  primary  units  in 
the  derived  unit.  It  is  clear  that  the  rational  processes 
of  parting  and  wholing  that  ultimately  give  clear  ideas 
of  tmit  and  number,  must  also  bring  out  clearly  the  re- 
lation between  these  two  factors  in  measured  quantity  : 
the  smaller  the  unit  the  larger  the  number  ;  or,  the  num- 
ber of  the  measuring  units  in  the  quantity  varies  in- 
versely as  the  number  of  primary  units  in  the  derived 
unit.  Measuring  a  lengtli  of  one  foot  by  a  6-incli  unit, 
by  a  3-inch  unit,  and  by  a  1-inch  unit,  the  numbers  are 
respectively  2,  4,  and  12  ;  measuring  a  lengtli  of  one 
decimetre  by  10,  20,  30,  40,  50  centimetres,  the  num- 
bers are  respectively  10,  5,  3J,  2^,  2  ;  measuring  $20 
by  the  $1  unit,  $2  unit,  $4  unit,  the  numbers  are  20, 10, 
5,  etc.  In  the  constructive  exercises  already  described, 
attention  to  measuring  unit  and  its  times  of  repetition 
must  lead  to  the  conscious  recognition  of  this  principle, 
which  is  fundamental  in  number  as  measurement.  It 
has  already  been  given  in  the  complete  statement  in 
"  fractional "  form  of  the  process  of  measurement : 
Any  'measured  quantity  may  be  expressed  in  the  form 

12  3   4  n 

75  o'  oi  7^  •  •  •  -•    This  principle  is,  of  course,  the  basal 

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THE  PSYCHOLOGY  OF  NUMBER. 


principle  in  the  treatment  of  fractions  (because  it  is  tlic 
primary  principle  of  number,  and  "  fractions  "  are  num- 
bers)— namely,  both  terms  of  a  fraction  may  be  multi- 
plied or  divided  by  the  same  number  without  altering 
the  value  of  the  fraction. 

The  Laiv  of  Coniinutation. — The  development  of 
this  principle  through  the  rational  use  of  its  idea — 
that  is  through  the  use  of  the  facts  supplied  by  sense- 
perception  in  the  rational  use  of  things — is  the  develop- 
ment of  the  psychological  law  of  commutation  which  is 
primary  and  essential  in  all  mathematics.  The  method 
that  ignores  this  necessary  relation  between  times  and 
parts,  or  regards  them  as  totally  different  things,  never 
leads  to  a  clear  conception  of  this  important  principle. 
It  is,  as  a  consequence,  always  iinding  difficulties  where, 
for  rational  method,  none  really  exist.  The  principle 
is  difficult  for  the  child  only  when  the  method  is  wrong. 
With  right  presentation  of  nuxterial  he  can  have  no  dif- 
ficulty in  seeing  that  the  larger  the  units  the  smaller 
their  number  in  a  given  (pumtity.  When  he  counts 
out  a  collection  of  24  objects  into  piles  of  3  each,  and 
into  piles  of  6  each,  can  he  fail  to  see  that  the  re- 
spective numbers  differ  ?  And  with  rightly  directed 
attention  to  the  concrete  processes,  may  he  not  be  led 
slowly,  perhaps,  but  surely,  to  a  clear  thought  of  how 
and  why  they  differ  ?  The  learner  can  not  help  seeing, 
for  example,  the  difference  between  the  2-inch  unit  and 
the  6-inch  unit,  and  the  corres})onding  difference  between 
six  times  and  three  times.  To  see  clearly  is  to  think 
clearly  ;  there  is  a  rationality  in  rationally  presented 
facts,  and  this  rationality  leads  with  certainty  to  a  com- 
plete recognition  of  the  meaning — the  law — of  the  facts. 


' 


ON  PRIMARY  NUMBER  TEACHING. 


109 


Should  he  Used  from  the  First. — Thus  the  idea  of 
the  law  of  coninuitation  can  be  used  from  the  first ; 
rather,  must  be  used  if  clear  and  adequate  ideas  of 
number  are  to  be  gained,  and  numerical  operations 
are  to  be  a  thing  of  meaning  and  of  interest.  As 
already  seen,  it  is  impossible  to  know  12  objects,  as 
measured  by  fours,  without  at  the  same  time  know- 
ing it  as  measured  by  threes.  So  it  is  in  the  actual 
operation  with  things  :  we  count  out  a  quantity  of  12 
things  into  groups  of  four  things  each,  and  find  the 
number  of  the  groups  to  be  three  y  and  we  count  out 
the  12  things  into  four  groups,  and  find  in  each  group 
three  things.  In  both  cases  the  operations  are  alike  ;  in 
neither  is  it  possible  to  get  the  result  without  using 
counts  of  four  things  each.  The  child  counts  out  12 
things  into  groups  of  4  things  each  ;  how  many  groups  ? 
lie  counts  them  out  into  4  groups ;  how  many  things 
in  each  group  ?  In  both  cases  he  sees  that  he  counts  by 
fours.  lie  counts  10  things  in  groups  of  5  things  each  ; 
how  many  groups  ?  He  counts  them  out  into  5  groups; 
how  many  in  each  group  ?  In  both  operations  he  sees 
that  he  counts  out  into  groups  of  five  things  each. 
Twenty  cents  are  to  be  equally  divided  among  5  boys; 
how  many  cents  will  each  boy  get  ?  Twenty  cents  are 
divided  equally  among  a  number  of  boys,  giving  each 
boy  5  cents  ;  how  many  boys  are  there  ?  The  child  per- 
forms the  operation  with  counters,  and  finds  in  both 
cases /6>z«/",  meaning  4  boys  in  the  one  case,  4  cents  in 
the  other ;  he  sees  that  in  both  examples  the  operation 
was  a  counting  hyfves,  and  he  will  soon  be  in  posses- 
sion of  the  important  truth — which  many  teachers,  and 
even  teachers  of  teachers,  seem  not  to  know — that  one 


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THE  PSYCHOLOGY  OF  NUMBER. 


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^y^/i  of  20  units  of  any  kind  is  four  units,  becavse 
Jive  units  must  be  repeated  four  times  to  measure  20 
units. 

There  will  be  but  little  difficulty  in  further  illus- 
trating the  principle  by  applying  it  to  exact  measure- 
ment. Three  4-inch  measures  make  up  the  linear  foot ; 
the  child  sees  that  one  inch  counted  out  of  each  of  the 
3  4-inch  units  gives  a  whole  of  3  inches  ;  another  inch 
thus  counted  out  gives  another  whole  of  3  inches,  etc. ; 
so  that  in  all  there  are  4  units  of  3  inches  each.  Since, 
however,  the  principle  is  more  distinctly  illustrated  by 
symmetrical  groupings  of  the  measuring  units  (see  page 
34),  the  separate  primary  units  niay  be  used  with  advan- 
tage— for  example,  for  the  4-inch  measure  use  the  four 
single  units  which  compose  it.  With  the  proper  use  of 
such  parting  and  wholing  exercises  the  child  can  not 
fail  to  comprehend  in  due  time  this  fundamental  prin- 
ciple in  number  and  numerical  operntions.  Seeing 
clearly  that  the  unit  is  composed  of  minor  units,  that 
it  is  repeated  to  make  up  or  measure  some  quantity,  he 
can  not  fail  to  see  that  each  and  all  of  its  minor  units 
are  repeated  the  same  number  of  times. 

What  has  been  joined  together  by  a  psychical  law 
can  not  be  divorced  without  checking  or  distorting 
mental  growth.  It  is  known  from  experience  that 
when  the  constructive  exercises  already  referred  to  are 
carried  on  under  rational  direction,  the  use  of  the  re- 
lated ideas  grows  natui'ally  and  surely  into  a  conscious 
recognition  of  the  relations  :  how  much  and  how  many, 
quantity  and  its  instrument  of  measurement,  parts  and 
times,  measuring  unit  and  measured  whole,  measuring 
minor  unit  and  its  measured  minor  whole,  correlated 


m 


ON  PRLMARY  NUMBER  TEACHING. 


ITl 


'  M 


factors  of  product,  correlated  processes  of  division,  are 
all  seen  in  their  true  logical  and  psvcholonical  inter- 
relation. These  things  are  organically  connected  by 
necessary  laws  of  thought ;  the  method  which  is  ra- 
tionalized by  this  idea  makes  arithmetic  a  delight  to 
the  pupil  and  a  powerful  educating  instrument.  A 
method  which  violates  this  necessarv  law  of  mind  in 
dealing  with  quantity — constantly  obstructing  the  origi- 
nal action  of  the  mind — makes  arithmetic  a  thine:  of 
rule  and  routine,  uninteresting  to  the  teacher,  and  prob- 
ably detested  by  the  learner. 

Ifake  Ilasie  slowli/. — As  ali-eady  suggested,  time  is 
necessary  for  the  completion  of  the  idea  of  nund)er. 
Under  sound  instruction  a  working  conce])tion  suitable 
for  a  primary  stage  of  development  may  be  readily  ac- 
quired, and  may  be  used  for  higher  development.  But 
a  perfectly  clear  and  delinite  conception  of  number  is 
a  ])roduct  of  growth  l)y  slow  degrees.  The  power  of 
numerical  abstraction  and  generalization  can  not  be  im- 
parted at  will  even  by  the  most  painstaking  teacher. 
Ilence  the  absurdity  of  making  minute  mechanical 
analysis  a  substitute  for  nature's  sure  but  patient  way. 
It  seems  to  be  thought  that  mechanical  drill  upon  a  few 
numbers — a  drill  which,  if  rational,  would  really  use 
ratio  and  proportion — will  in  some  unexplained  way 
"  impart "  the  idea  of  number  to  the  child  apart  from 
the  self-activity  of  which  alone  it  is  the  product.  And 
this  fallacious  idea  is  strenc^thened  by  the  fluent  chatter 
of  the  child — the  apt  repeater  of  mere  sense  facts — 
about  the  "  equal  numbers  in  a  number,"  the  "  equal 
numbers  that  make  a  number,"  and  all  the  rest  of  it. 
This  routine  analysis  and  parrotlike  expression  of  it  are 


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11 


172 


THE  PSYCHOLOGY  OF  NUMBER. 


a  direct  violation  of  the  psychology  of  number.     The 
idea  of  number  can  not  be  got  from  this  forcing  process. 
The  conscious  grasp  of  the  idea,  we  repeat,  must  come 
from  rational  use  of  the  idea,  and  is  all  but  impossible 
by  monotonous  analyses  with  a  few  "simple"  numbers  ; 
it  is  absolutely  impossible  in   the   immature  state  of 
the  child-mind  in  which  it  is  attempted.      The  child 
must,  once  more,  freely  and  rationally  use  the  ideas ; 
must  operate  with  many  things,  using  many  numbers, 
before  the  idea  of  number  can  possibly  be  developed. 
We  are  omitting  the  things  the  child  can  do — rationally 
iise  numerical  ideas — and  forcing  upon  him  things  that 
he  can  not  do — form  at  once  a  complete  conception  of 
number  and  numerical  relations.      It  is  high  time  to 
change  all  this :  to  omit  the  things  he  can  not  do,  and 
interest  him  in  the  things  he  can  do.     In  the  compara- 
tively formal  and  mechanical  stage  tliere  must  be  a  cer- 
tain amount  of  mechanical  drill — mechanical,  yet  in  no 
small  degree  disciplinary,  because  it  works  with  ideas 
which,  though  imperfect,  are  adequate  to  the  stage  of 
development  attained,  and  through  rational  use  become 
in   due   time  accurate  scientific  conceptions.      Besides 
valuable  discipline,  the  child  gets  possession  of  facts 
and   principles — of  elementary  knowlc^^e,  it  may  be 
said — which  are  essential  in  his  progress  towards  sci- 
entific concepts  and  organized  knowledge.      It  seems 
absurd,  or  worse  than  absurd,  to  insist  on  thoroughness, 
on  perfect  number  concepts,  at  a  time  when  perfection 
is  impossible,  and  to  ignore  the  conditions  under  which 
alone  perfect  concepts,  can  arise — the  wise  working  with 
imperfect  ideas  till  in  good  time,  under  the  law  con- 
necting idea  and  action,  facile  doing  may  result  in  per- 


ON   PRIMARY  NUMBER  TEACHING. 


173 


feet  knowing.  Following  the  nonpsycliological  method 
liinders  the  luitural  action  of  the  mind,  and  fails  to  pre- 
pare the  child  for  subsequent  and  higher  work  in  arith- 
metic. The  rational  method,  promoting  the  natural  ac- 
tion of  the  mind  by  constructive  processes  which  use 
number,  leads  surely  and  economically  to  clear  and 
definite  ideas  of  number,  and  thoroughly  prepares  for 
real  and  rapid  progress  in  the  higher  work. 

The  Stat'timj  Point. — It  is  commonly  assumed  that 
the  child  is  familiar  with  a  few  of  the  smaller  numbers 
— with  at  least  the  number  three.  He  has  undoubtedly 
acquired  some  vague  ideas  of  number,  because  he  has 
been  acting  under  the  number  instinct ;  he  has  been 
counting  and  measuring.  But  he  does  not,  because  he 
can  not,  knoio  the  number,  lie  knows  3  things,  and  5 
or  more  things,  when  he  sees  them  ;  he  knows  that  5 
apples  are  more  than  3  apples,  and  3  apples  less  than  5 
apples.  But  he  does  not  know  three  in  the  mathemat- 
ical or  psychological  sense  as  denoting  measurement  of 
quantity — the  repetition  of  a  unit  of  measure  to  equal 
or  make  up  a  magnitude — the  ratio  of  the  magnitude  to 
the  unit  of  measure.  If  he  does  know  the  number  3, 
in  the  strict  sense,  it  is  positively  cruel  to  keep  him 
drilling  for  months  and  months  upon  the  number  five 
and  "  all  that  can  be  done  with  it,"  and  years  upon  the 
mimber  twentv. 

The  Nxmiljer  Two. — There  can  be  little  doubt  that 
among  the  early  and  imperfect  ideas  of  number  the 
idea  of  two  is  first  to  appear.  From  the  first  vague 
feeling  of  a  this  and  a  that  through  all  stages  of  growth 
to  the  complete  mathematical  idea  of  two,  his  sense  ex- 
periences are  rich  in  twos :  two  eyes,  two  ears,  two 


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174 


THE  PSYCHOLOGY  OF  NUMBER. 


hands,  this  side  and  that,  up  and  down,  riglit  and  left, 
etc.  The  whole  structure  of  things,  so  to  speak,  seems 
to  abound  in  twos.  But  it  is  not  to  be  supposed  that 
this  common  experience  has  given  him  the  number  two 
as  expressing  order  or  relation  of  measuring  nnits.  The 
two  things  which  he  knows  are  qualitative  ones-,  not 
units.  Two  is  not  recognised  as  expressing  the  same 
relation,  however  the  units  may  vary  in  quality  or  mag- 
nitude ;  it  is  not  yet  one  +  one,  or  one  taken  two  times 
— two  apples,  tivo  5-apples,  two  10-apples,  two  100-apples  ; 
or  tivo  1-inch,  two  5-inch,  two  10-incli,  two  100-inch  units ; 
in  short,  o?ie  unit  of  measure  of  any  quantity  and  any 
value  taken  tivo  times.  But  his  large  experience  with 
pairs  of  things,  and  the  imperfect  idea  of  two  that  neces- 
sarily comes  iirst,  prepare  him  for  the  ready  use  of  the 
idea,  and  the  comparatively  easy  development  of  it. 
There  must  be  a  test  of  how  far,  or  to  what  extent,  he 
knows  the  number  two.  This  is  supplied  by  construct- 
ive exercises  with  things  in  which  the  idea  of  two  is 
prominent.  The  child  separates  a  lot  of  beans  (say  8) 
into  two  equal  parts,  and  names  the  number  of  the  parts 
two  /  separates  each  part  into  two  equal  parts,  and  names 
the  number  of  the  parts  tivo  /  separates  each  of  these 
parts  into  two  equal  parts,  and  names  the  number  of  the 
single  things  tivo.     Or,  arranging  in  perceptive  forms, 

how  many  ones  in     ?     How  many  twos  in  ?     How 


many  pairs  of  twos  in 


•      0 


?     Similar  exercises 


and  questions  may  be  given  with  splints  formed  into 
two  squares,  and  into  two  groups  of  two  pickets  each ; 
with  12  splints  formed  into  two  squares  with  diagonals 
(see  page  106) ;  then  each  square  (group)  formed  into 


ON   PRLMARY  NUMBER  TEACHING. 


175 


two  triangles ;  how  many  squares  ?  how  many  ti'iangles 
(unit  groups)?  how  many  pairs  of  triangles?  Exact 
measurements  are  to  accompany  such  exercises  :  the 
12-inch  length  measured  by  the  6-inch,  this  again  by 
the  3-inch  ;  how  many  6-inch  units  in  the  whole  ?  3-inch 
units  in  the  6-inch  units  ?  pairs  of  3-inch  units  in  the 
whole  ?  Put  1-inch  units  together  to  make  the  2-inch 
unit,  the  2  inch  units  to  make  the  4-inch  unit,  two  of 
these  to  make  8  inches,  etc.  It  is  not  meant  that  these 
exercises  are  to  be  continued  till  the  number  two  is 
thoroughly  mastered ;  they  carry  with  them  notions  of 
higher  numbers,  without  which  a  conception  of  two  can 
not  be  reached.  Beware  of  "  thoroughness  "  at  a  stage 
when  thoroughness  (in  the  sense  of  complete  mastery) 
is  impossible. 

The  Number  Three. — The  number  three  is  a  much 
more  difficult  idea  for  the  child.  As  in  the  case  of  two, 
he  knows  three  objects  as  more  than  two  and  less  than 
four ;  three  units  in  exact  measurement  as  more  than 
two,  etc.  But  he  does  not  know  three  in  the  strictly 
numerical  sense.  He  may  know  tw^o  fairly  well — as 
a  working  notion — without  having  a  clear  idea  of  the 
ordered  or  related  ones  making  a  whole.  In  three,  the 
ordering  or  relating  idea  must  be  consciously  present. 
It  is  not  enough  to  see  the  three  discriminated  ones ; 
they  must  at  the  same  time  be  related^  unified — a  first 
one,  a  second  one,  a  third  one,  three  07ies^  a  one  of  three. 
Three  must  be  three  units — measuring  parts  of  a  quali- 
tative whole — units  made  up,  it  may  be,  of  2,  or  3,  or 
4  ...  or  71  minor  units.  If  12  objects  are  counted  off 
in  unit-groups  of  4  each,  15  objects  into  unit-groups 
of  5  each,  18  objects  into  unit-groups  of  6  each,  30  ob- 
13 


; 

1'    ; 

.1    , 

1 1 
1  ■• 

m  i 

ii 

•'I 

!( 

h 


u 


i: 


176 


THE  PSYCHOLOGY  OF  NUMBER. 


jects  into  uiiit-groiips  of  10  each,  the  number  in  each 
and  every  case  ninst  be  recognised  as  three.  The  num- 
ber three,  in  fact,  may  be  taken  as  the  test  of  progress 
towards  the  true  idea  of  number,  and  of  the  child's  abil- 
ity to  proceed  rapidly  to  higher  numbers  and  nunierical 
relations.  If  he  knows  three,  if  he  has  even  an  intelli- 
gent working  conception  of  three,  he  can  proceed  in  a 
few  lessons  to  the  number  ten,  and  will  thus  have  all 
higher  numbers  within  comparatively  easy  reach. 

The  child  is  not  to  be  kept  drilling  on  the  number 
thi-ee  until  it  is  fully  mastered.  "What  was  said  of  two 
applies  equally  to  three  :  it  can  not  be  mastered  without 
the  implicit  use  of  higher  numbers.  But,  while  dealing 
with  higher  numbers,  three  may  be  kept  in  view  as  the 
crucial  point  in  the  development  of  exact  ideas  of  num- 
ber. There  are  to  be  constructive  exercises  with  vari- 
ous measuring  units  in  which  the  threes  are  prominent. 
In  getting  the  number  two,  there  has  been  practically 
a  use  of  three  ;  for  where  two  is  pretty  clearly  in  mind, 
four  is  not  far  behind  it  in  definiteness.  When,  for 
example,  it  is  clearly  seen  that  two  2-inch  units  make 
up  the  4-inch  unit,  and  two  4-inch  units  make  up  8 
inches,  the  two  2-inch  units  are  perceived  as  four — i.  e., 


and 


are  seen  as 


.  But  for  the  complete  con- 
ception of  J'our  it  nmst  be  related  not  only  to  two 
(   \  but  also  to  three  ;  in  other  words,  there  must 

be  rational  counting — we  must  pass  througli  the  mim- 
her  three  to  complete  recognition  of  the  number  four. 

But,  as  already  suggested,  there  are  to  be  special 
exercises  in  threes.  The  dozen  splints  are  to  be  used 
in  constructing  squares  and    triangles.      How  many 


.  f 


ON   PRIMARY   NUMBER  TEACHING. 


177 


squares?  IIow  many  splints  needed  to  make  one  tri- 
angle ?  Each  of  the  two  6-splints  is  to  be  made  into 
as  many  triangles  as  possible.  How  many  triangles  in 
each  group  ?  How  many  in  all  ?  "  Two  twos,  or  three 
triangles  and  one  more."  Each  of  the  twf>  6-splints  is 
to  be  made  into  as  many  pickets  (A)  as  possible.  How 
many  pickets  in  each  group  ?  Hc>w  many  in  all  ?  "  Two 
3-pickets."  In  the  two  3-pickets  how  many  2-pickets  ? 
Thus,  also,  with  exact  measurements.  The  4-inch  units 
in  the  foot,  the  3-inch  units  in  the  foot  and  in  the  9- 
inch  measure,  the  2-incli  units  in  the  6-inch  measure ; 
the  number  of  3  square-inch  units  in  the  3-inch  square, 
the  number  of  4-inch  units  in  the  3-inch  by  4-inch  rec- 
tangle, the  number  of  3-centimetre  units  in  the  9  cen- 
timetres, etc. 

Other  Numhers  to  Ten, — Just  as  when  the  child 
has  a  good  idea  of  two  he  implicitly  knows  four,  so 
when  he  has  a  good  idea  of  three  he  has  a  fair  idea  of 

six  as  two  threes.     In  (symbolizing  any  units 

whatever)  the  two  3-units  are  jperceived  as  six  units. 
This  perception  is  connected  with  1,  2,  3,  4,  of  which 
the  child  has  already  good  working  ideas,  and  has  only 
to  be  related  with  the  number  five  in  order  to  fix  its 
true  place  in  the  sequence  of  related  actf3,  one,  two  .  .  . 
six,  which  completes  the  measurement.  In  this  we  pass 
through  five ;  five  is  the  connecting  link  between  four 
and  six  in  the  completed  sequence.     Attention  to  the 

perception  of  six  units  in  discriminates  the  five 


•  •   • 

•  •    • 


units 


•    •   • 


from  both  the  four  and  the  six,  and  con- 


1 1 

I 


M 


ceives  their  proper  relation  to  the  four  units  whicli  are 


f 


.1"    * 

1.1), 

1  ji 

1  M  ■   '' 

,!•>    ' 

^ '  i:  i 

^:;U 

^i!= }: ; 

:\.\- 

&.  ■ 


^:i'^'' 


178 


THE  PSYCHOLOGY  OF  NUMBER. 


•    • 


a  part  of  tliein,  and  to  the  six  units  of  which  they  are 
a  part.     When  there  is  a  fair  idea  of  four  it  is  an  easy 

step  to  a  fair  idea  of  eight ;  4  +  4  =  8. 

is  not  much  more  difficult  than  2  +  2  =  4 — 

From  the  examination  of  eight  comes  the  perception 
of  seven,  and  a  conception  of  its  relation  as  one  more 

than  seven  and  one  less  than  eiorht.     From  five,     •    ? 
it  is  by  no  means  a  difficult  step  to  ten,  as  two  lives, 
•  •    ;  and  a  comparison  of  this  with  eight  read- 

ily leads  to  the  perception  of  nine,     •  ,  and  to 

a  conception  of  its  relation  to  both  eight  and  ten. 

Is  Ten  twice  Five? — From  the  error  of  consider- 
ing the  unit  as  a  fixed  thing,  and  number  as  arising 
from  aggregating  one  by  one  other  isolated  things, 
arises  apparently  the  fallacious  idea  that  to  master 
ten,  for  instance,  is  twice  as  difficult  a  task  as  to  mas- 
ter five.  Hence  the  prevailing  practice  of  devoting  six 
months  of  precious  school  life  in  wearying  and  repul- 
sive "  analyses  of  the  number  five,"  and  finding  "  a  year 
all  too  short "  for  similar  analyses  of  the  number  tenT 
But  it  must  be  clear  that  by  proper  application  of  the 
measuring  idea,  wisely  directed  exercises  in  parting  and 
wholing  which  promote  the  original  activity  of  the  mind 
in  dealing  with  quantity,  it  is  easy  and  pleasant  to  get 
elementary  conceptions  of  number  in  general  in  the 
time  now  given  to  barren  grinds  on  the  number  ten ; 
not  scientific  conceptions,  indeed,  but  sufficiently  clear 
working  conceptions,  capable  of  large  and  free  appli- 
cations in  the  measurement  of  quantity — applications 


■'I 


ON   PRIMARY  NUMBER  TEACHING. 


179 


' 


which  alone  can  make  the  vague  definite,  and  at  last 
evolve  a  perfect  conception  of  number  from  its  first  and 
necessarily  crude  beginnings  in  sense-percej)tion. 

The  /Sequence  m  Prhnart/  Number  Teachlmj. — 
The  measuring  exercises  suggested,  operations  of  sepa- 
rating a  whole  into  equal  parts  and  remaking  the  whole 
from  the  parts,  correspond  to  the  processes  of  multipli- 
cation and  division.  But  they  are  not  these  processes, 
though  they  are  presentations  of  sense  facts  which  help 
the  unconscious  growth  towards  the  conscious  recogni- 
tion of  these  processes  as  phases  in  the  development  of 
the  measuring  idea.  It  has  been  shown  that  addition  and 
subtraction,  multiplication  and  division,  differ  in  psy- 
chological complexity  ;  that  eacli  operation  in  the  order 
named  makes  a  severer  demand  on  attention  ;  and  that, 
therefore,  while  operations  corresponding  to  the  higher 
processes  may — indeed,  must — with  discretion  be  used 
from  the  first,  they  should  not  be  made  the  object  of 
conscious  or  analytic  attention.  We  separate  a  quan- 
tity into  parts — the  presentative  element  in  division ; 
we  put  the  parts  together  again — the  presentative  ele- 
ment in  multiplication.  But  the  ideas  of  subtraction 
are  involved  in  these  operations — because  division  and 
multiplication  are  involved — and  as  demanding  less 
power  of  discrimination  and  relation  are  the  first  to 
be  analytically  taught.  The  separation  into  parts  is 
not  enough  ;  the  mere  putting  together  of  the  parts  is 
not  enough.  There  must  be  a  counting  of  the  parts 
making  up  the  whole:  one,  two,  three  .  .  .;  and  this 
means  addition  by  ones.  We  may  separate  any  12-unit 
quantity  into  equal  2-unit  parts ;  we  do  not  know  that 
there  are  six  parts  till  we  have  counted  the  parts,  one, 


I 


I 


-  ! 

J                  ( 
J  ■               ■ 

•!l!  : :: 

^f'( 

« '  » 

i.  i 

h  ■ 

1  ■  .' 

"-■''111     ' 

X^*\  1 

•!,'K«; 

i! 

^H!: 


;!!i^ 


■!  , 


II: 


i 


I 


II 

liii 


( / 


180 


THE  PSYCHOLOGY  OF  NUMBER. 


two  .  .  .  six — tliat  is,  a  Jlrst  one,  a  second  one  .  .  ., 
six  ones. 

Nor  will  the  most  ingenious  presentations  of  sense 
material  free  us  from  this  fundamental  process.  Such 
presentations  help  to  make  the  process  rational.;  they 
do  not  supersede  it.  If  a  good  working  idea  of  four 
has  been  got,  it  carries  with  it  the  idea  of  three.  But 
it  does  not  follow  that  seven  (4  H-  3)  is  known  without 


includes  the  presenta- 


does 


counting.     The  presentation 

tion  ;  but  the  mere  perception  of  • 

not  give  the  number  seven.  We  perceive  the  four  and 
the  three,  and  we  know  these  numbers  because  we  have 
previously  analyzed  each  of  them,  and  put  its  units  in 
ordered  relation  to  one  another — that  is,  counted  them. 
We  have  a  perception  of  the  group  which  represents 
the  union  of  these  two  numbers,  but  we  do  not  really 
know  seven  till  we  put  the  three  additional  units  in 
ordered  relation  to  the  four;  or,  in  other  words,  till 
(starting  with  four)  we  count  five,  six,  seven,  thus  fix- 
ing the  places  of  the  new  units  in  the  sequence  of  acts 
by  which  the  whole  is  measured.  We  are  not  to  rest 
satisfied,  on  the  one  hand,  with  mechanical  counting — 
mere  naming  of  numbers — nor,  on  the  other  liand,  with 
mere  perceptions  of  units  unrelated  by  counting.  This 
consciously  relating  process  gives  the  ordinal  element 
in  number ;  counting,  for  example,  the  units  in  a  seven- 
unit  quantity,  when  we  reach  four  we  must  recognize 
that  four  is  the  fourth  in  the  sequence  of  acts  by  which 
the  whole  is  constructed. 

The  following  points  in  the  development  of  number 
seem,  therefore,  perfectly  clear  : 


A 


A 


lis 


ON  PRIMARY  NUMBER  TEACHING. 


181 


1.  The  measuring  idea  should  be  made  prominent 
by  constructive  exercises  such  as  have  been  suggested. 

2.  There  must  be  rational  eountinir — relating — of 
the  units  of  measurement.  This  is  addition  by  ones. 
It  is  impossible  to  know,  for  example,  ten  times,  with- 
out having  added  ten  ones. 

3.  While  use  may — rather  should — be  made  of  the 
ratio  idea  (division  and  multiplication),  the  mastery  of 
the  combinations  to  ten  should  be  kept  chiefly  in  view ; 
that  is,  addition  and  subtraction,  with  the  emphasis 
on  addition,  should  be  first  in  attention,  but  with  exer- 
cises in  the  higher  processes.  This  7mist  be  the  course 
if  the  ideas  of  unit  and  number  are  to  be  rationally 
evolved.  In  counting  up,  for  example,  the  four  3-inch 
units  in  the  foot  measure,  the  child  first  feels  and  at  last 
sees  that  one  unit  is  1  out  of  four  ;  two,  2  out  of  four  ; 
three,  3  out  of  four ;  or  1-fourth,  2-fourths,  3-fourths  of 
the  whole. 

4.  To  help  in  this  relating  as  well  as  in  the  discrim- 
inating process,  rhythmic  arrangements  of  the  actual 
units,  and  of  points  or  other  symbols  of  them,  may  be 
used  with  remarkable  effect.  The  real  meaning  of  five, 
as  denoting  related  units  of  measure,  w411  clearly  and 
quickly  be  seen  when  the  units  (or  their  symbols)  are 

arranged  in  the  form 


and  so  with  other  num- 


bers. The  results  of  the  entire  mental  operation  of 
analysis-synthesis,  by  which  the  vague  whole  has  been 
made  definite,  are  given  in  these  perceptive  forms. 

5.  Hence,  during  the  first  year  at  school  we  need 
not  confine  our  instruction  to  5,  or  50,  or  500,  if  we 
follow  rational  methods.     The  first  thing,  then,  is  to 


m 


m 


I 


i 


WT 


182 


THE  PSYCHOLOGY  OF  NUjMBER. 


M'i 


{' 


uiii  "« 


::  I 


i|' 


it; 


.1   ' 


see  that  the  child  gains  not  a  thorough  mastery  but 
a  good  working  idea  of  the  number  ten.*  After 
exercises  in  parting  and  wholing,  some  notions  of 
unit  and  number  have  been  gained,  and  a  formal 
start  is  made  for  ten^  the  instrument  of  instruments 
in  the  development  of  number.  Many  a  12-unit  quan- 
tity has  been  divided  into  equal  parts.  Working 
ideas  of  the  numbers  from  one  to  six  have  been  ob- 
tained ;  the  child  works  with  them  to  make  the  num- 
bers one  to  six  more  definite.  If  the  preliminary  con- 
structive operations  have  been  w^isely  directed  the  task 
is  now  an  easy  one.  Not  long-continued  and  monoto- 
nous drill  on  all  that  can  be  done  upon  the  number 
ten  is  needed,  but  a  little  systematic  work,  with  the 
ideas  which  from  free  and  spontaneous  use  are  ready  to 
flash,  as  it  were,  into  conscious  recognition.     Using  the 

number  forms  for  six,  ,  it  will  need  but  few  ex- 

•   •  • 

ercises  to  make  perfectly  clear  the  real  meaning  of  six, 
and  then  the  remaining  numbers  Y  ...  10  are,  as  means 
for  further  progress,  within  easy  reach.  In  using  these 
number  percepts  the  "  picturing  "  power  should  be  cul- 
tivated.    Five,  six,  seven   as  five  and  two,  eight  as  two 

*  For  the  constructive  exercises  referred  to,  various  objects  and 
measured  things  may  be  used  as  counters;  but  since  exact  ideas  of 
numbers  can  arise  only  from  exact  measurements,  and  since  ten  is 
the  base  of  our  system  of  notation,  the  metric  system  can  be  used 
with  great  advantage.  The  cubic  centimetre  (block  of  wood)  may 
be  taken  as  a  primary  unit  of  measure  ;  a  rectangular  prism  (a  deci- 
metre in  length),  equal  to  ten  of  these  units,  will  be  the  10-unit,  and 
ten  of  these  the  100-unit.  The  units  may  be  of  different  colours, 
and  the  units  of  the  decimetre  alternatelv  black  and  white.  A  foot- 
rule,  with  one  edge  graduated  according  to  the  English  scale,  and 
the  other  according  to  the  metric  scale,  is  a  most  useful  help. 


If 


ON  PRIMARY  NUMBER  TEACHING. 


183 


fours,  ten  as  two  fives,  and  five  twos,  etc.,  should  be 
instantly  recognised.  In  exercises  upon  the  combina- 
tions of  six  we  have  the  whole  and  all  the  related  parts 
distinctly  imaged.  The  analysis  of  the  visual  forms 
has  been  made  5  +  1  =  6,  4  +  2^6,  etc.  Now  cover 
the  5  ;  how  many  are  hidden  ?  how  many  seen  ?  Cover 
the  four ;  how  many  are  hidden  ?  liow  many  seen  ?  And 
so  with  all  the  combinations,  taking  care  that  the  related 
pairs  of  number  are  seen  as  related — for  example,  6  =  5 
-f-  1  =  1  -f-  5.  This  insures  a  repetition  of  the  number 
activity  in  each  case,  and  the  ability  to  recognise,  on 
the  instant,  any  number  ;  to  see  not  only  a  whole  made 
lip  of  parts,  but  also  the  definite  numher  of  parts  in  the 
whole. 

Comlnnations  of  the  10  -  Units. — It  is  hardly  neces- 
sary to  say  that,  with  a  fair  degree  of  expertness  (not  a 
perfect  mastery)  in  handling  these  twenty -five  primary 
combinations,  rapid  progress  may  be  made  in  the  use 
of  the  higher  numbers.  As  soon  as  a  good  idea  of  ten 
is  gained,  the  pupil  handles  the  tens,  using  as  the  ten- 
unit  the  decimetre  already  described  (or  other  convenient 
measures),  and  going  on  to  10-tens,  even  more  easily 
and  pleasantly  than  he  proceeded  to  10  "ones."  Is 
there  any  known  law,  save  the  law  of  an  utterly  irra- 
tional method,  that  confines  the  child  for  six  months  to 
the  number  five,  for  twelve  months  to  the  number  ten, 
for  another  year  to  the  number  twenty,  actually  exact- 
ing two  whole  years  of  the  child's  school  life — to  isay 
nothing  of  the  kindergarten  period — before  letting  him 
attempt  anything  with  the  mysterious  thirty  ?  If  a 
child  has  really  learned  that  five  and  three  are  eight, 
does  he  not  know  that  5  inches  and  3  inches  are  8  inches, 


m 


s\m 


m 


'\m' 


W 


mr 


III  r  - 

i'i  J'  r 


Sfci 


i!Hr 


■I  - 
I ' ; !  ■ 

.'i: 

1! 
■'I 

H 


Ihl: 


Nl^t:n 


■ .  r 


184: 


THE  PSYCHOLOGY  OF  NUMBER. 


5  feet  and  3  feet  are  8  feet,  5  yards  and  3  yards  are  8 
yards,  5  miles  and  3  miles  are  8  miles  ?  Do  live  and 
three  cease  to  be  eight,  or  ten  repudiate  5  times  two 
when  the  unit  of  measure  is  changed  ?  As  a  matter  of 
fact,  when  the  child,  under  rational  instruction,  gets  a 
good  grip  of  three  he  quickly  seizes  ten,  the  master  key 
to  number.  As  soon  as  he  comes  to  ten  let  him  for- 
mally practise  with  the  tens  the  combinations  he  has 
learned  with  ones :  the  ten  is  a  unit,  because  it  is  to  be 
repeated  a  number  of  times  to  make  up  another  quan- 
tity, just  as  much  as  "  one  "  is  a  unit  because  it  is  used 
a  number  of  times  to  make  up  a  quantity.  When  it  is 
known  that  live  units  and  three  units  are  eight  units,  it 
is  known  for  all  units  of  measure  whatever.  There  is 
absolutely  no  limitation  save  this,  that  the  child  should 
have  a  reasonably  good  working  idea  of  the  unit  of 
measure.  If  he  knows,  for  instance,  that  5  feet  and  3 
feet  are  8  feet,  he  not  only  knows  that  5  miles  and  3 
miles  are  8  miles,  but  also  has  a  good  idea  of  the  dis- 
tance 8  miles,  provided  he  has  a  good  idea  of  the  unit 
mile.  So,  when  he  has  a  working  idea  of  a  10-unit 
quantity  as  compared  with  the  1-unit  quantity  which 
measures  it,  he  passes  with  the  greatest  ease  to  a  good 
idea  of  a  magnitude  measured  by  ten  such  10-unit 
quantities.  In  using  the  mr'  'C  units,  previously  re- 
ferred to,  he  has  analyzed  the  c^oimetre  prism,  has  com- 
pared it  with  the  minor  unit — the  cubic  centimetre — has 
found  that  it  takes  ten  of  these  minor  units,  or  one  of 
1  taken  ten  times,  to  equal  it,  etc.  He  goes  through 
J..Ki  same  constructive  process  with  ten  of  these  10-unit 
measures ;  he  puts  ten  of  them  together  to  make  a 
square  centimetre  ;  he  analyzes  and  compares  ;  he  uses 


ON  PRIMARY  NUMBER  TEACHING. 


185 


li^ 


this  new  unit  measure  just  as  he  used  the  minor-unit 
measure,  the  single  cube  ;  he  finds  that  5  units  and  3 
units  are  8  units,  6  units  and  2  units  are  8  units,  .  .  . 
7  units  and  3  units  are  10  units,  8  units  and  2  units 
are  10  units ;  and  all  the  rest  of  it.  He  finds  also, 
just  as  certainly  as  lie  found  in  operating  with  the  10 
cubes,  that  to  make  up  the  whole  he  is  measuring,  one 
10-unit  has  to  be  repeated  ten  times,  2  ten-units  five 
times,  and  5  ten-units  two  times.  In  fine,  he  knows  as 
much  about  the  whole,  the  ten  10-units,  the  100  minor 
units,  as  he  knows  about  the  1  decimetre,  the  10  minor 
units  ;  knows  it  just  as  well,  is  just  as  clear  as  to  the  re- 
lations of  the  several  parts ;  for  he  has  analyzed  the 
undefined — the  unknown — down  to  its  known  constit- 
uf"  i,  and  built  it  up  again  l)y  known  relating  pro- 
poses. He  does  not  perceive  it  just  so  well,  does  not 
image  it  as  a  quantity  of  a  hundred  parts  ;  but,  never- 
theless, he  has  a  fairly  definite  conception  of  the  quan- 
tity as  a  measared  whole. 

Naming  the  Numhers. — in  counting  the  tens  the 
names  of  the  new  numbers  may  be  given ;  or,  rather,  a 
name  or  two  may  be  given,  and  the  child  will  discover 
the  others  for  himself :  1  ten,  2  tens,  3  tens,  etc. ;  for 
two  tens  we  have  the  somewhat  special  name  twen-^y 
(twain-ten),  and  for  three  tens  ihiv-ty  i  what  name  for 
four  tens  ?  iovjive  tens  ?  Let  the  pupils  experience  as 
often  as  possible  the  joy  of  discovering  something  for 
themselves. 

While  the  work  with  tens  is  going  on,  practice  may 
be  had  in  the  analysis  of  two  tens,  so  as  to  lead  to  count- 
ing and  naming  the  numbers  from  ten  to  twenty,  twenty 
to  thirty,  etc.     Here,  again,  the  children  will  do  some- 


il 


■a] 


i 


w 


';.« 


i  1  > '   <! 


i|l 


mi 
.1" 


-    K 

r 


•i  H  '. 


H-  i 


'^•;l 


riii 


.1,  ,t 


186 


THE  PSYCHOLOGY  OF  NUMBER. 


thing  for  themselves.  The  counters  will  he  arranged, 
as  hefore,  so  as  to  facilitate  the  perception  of  the  corre- 
sponding numhers ;  12  will  he  recognised  as  two  O's,  as 
three  4's,  as  2  more  than  10,  and  so  on.  Eleven  (1  ten 
and  1)  and  twelve  (1  ten  and  2)  have  special  names. 
One  ten  and  3  is  named  thirteen  (=  three-teen).  What 
is  the  name  for  one  ten  and  four  ?  One  ten  and  five  ? 
One  ten  and  six  ?  The  comhinations,  the  part  relations, 
will  he  learned  through  intuitions  supplied  hy  the  meas- 
ured counters,  just  as  the  comhinations  of  ten  were 
learned.  The  first  learned  group  of  comhinations — 
those  of  ten — are  used  for  the  easy  mastery  of  the  second 
group,  those  from  11  to  19  inclusive.    In  ten  there  are : 

1.1.1      2      1     2      123      123     123 

1'    2'    3'    2'   4'    3'    5'  4'  3'    6'  5'  4'   T'  C   5 


4 
4' 


1 

8 


2 

7' 


3     4     1 

6'   5'    9 


2 

8 


3 

7' 


4     5 
,  p..     These  twenty-five 


6'  5* 


comhinations,  gained  through  intuitions  of  things  in  the 

way  already  descrihed,  and  applied  also  in  constructive 

processes  with  the  ten-units,  render  the  mastery  of  the 

remaining  twenty-Jive  comhinations  (11-20)  a  compara- 

..     ,  ,    ,       rr^  2     3     4     5.3456. 

tively  easy  task.     They  are :  ^,  8'  7'  6'  9'  8'  7'  6' 


4 

9' 


5     6     5 

8'  7'  9 


6 

8 


7'  ) 


7.  7 
8'  9' 


o  ;  Q .    These  fifty  com- 


hinations are  the  foundation  of  all  arithmetical  opera- 
tions, and  the  intelligent  mastery  of  them  should  from 
the  first  be  the  teacher's  guide  in  all  the  psychological 
constructive  processes  already  described  or  referred  to. 
The  dotation  of  the  Numhers. — The  figure  should 
follow  the  word  as  the  word  the  idea  of  the  number. 
"When  a  child  is  able  to  tell  the  number  of  the  units  in 


ill 


ON  PRIMARY  NUMBER  TEACHING. 


187 


any  measured  whole,  he  can  use  the  figure  that  denotes 
the  number.  The  early  use  of  " figures"  is  not  tlie 
cause  of  the  baldly  mechanical  ''  method  of  symbols  "  ; 
the  figures  are  not  responsible  for  the  machinelike 
movements,  any  more  than  "  words "  are  responsible 
for  the  machinelike  movements  in  mere  rote  learn- 
ing. In  both  cases  the  method  is  at  fault.  Words 
are  taught  not  as  words,  but  as  mere  sounds  signify- 
ing nothing;  so  "figures"  have  been  taught  as  empty 
signs,  with  which  certain  mystic  operations  may  be  per- 
formed. As  in  the  one  case  the  true  method  is  not 
words  without  things  nor  things  without  words,  but 
words  icith  things  ;  so  in  the  other,  it  is  not  numher 
without  figures  or  figures  without  number,  but  number 
with  figures.  The  intellectual  peril  comes  in  both  cases 
from  a  method  that  retards  and  warps  the  spontaneous 
action  of  the  mind.  In  getting  the  notation  of  ten 
and  the  higher  numbers,  it  is  almost  a  misuse  of  lan- 
guage to  say  that  there  is  real  difficulty.  Elaborate 
analyses  of  the  "  method  of  teaching  the  figure  6,"  and 
the  countless  inane  devices  for  teaching  the  notation 
of  the  numbers  ten  to  twenty,  and  twenty  to  one  hun- 
dred, should  have  no  place,  and  have  no  place,  in  rational 
method.  The  figures  from  zero  to  9  have  to  be  taught, 
given  authoritatively  in  connection  with  the  ideas  they 
denote.  The  symbol  for  ten  is  similarly  given  when  Xqh 
is  reached,  and  when  the  handling  of  the  10-units  begins. 
Show  the  10-unit  alone — that  is,  one  ten  and  no  units, 
and  state  that  it  is  expressed  thus,  10.  Then  express 
two  tens  and  no  units ;  three  tens  and  no  units.  The 
children  express  the  entire  series  up  to  a  hundred  with 
the  greatest  ease.     A  like  course  may  be  followed  for 


if 


SI 


ff 


\^ 

Ti 


:  ^ 


ill!* 

1r 


Hi 


I 
[  I  ■ 


ri 


188 


THE  PSYCHOLOGY  OF  NUMBER. 


the  imnibers  11  to  19.  This  symbol  (10)  denotes  one 
ten  and  no  units.  What  will  denote  one  ten  and  one 
unit?  one  ten  and  two  units?  Exactly  the  same  course 
may  be  followed  witli  all  the  higher  numbers. 

The  Hundred  Table. — Thus,  with  very  little  help, 
the  pupil  will  name  and  write  down  all  the  numbers 
from  1  to  100.  lie  will  be  greatly  interested  in  con- 
structing a  table  of  such  numbers  and  noticing  how 
they  are  formed.     The  first  column  on  the  left  has  to 

be  given  him  as 


0 

10 

20 

30 

40 

50 

60 

70 

80 

1 

11 

21 

31 

41 

51 

61 

71 

81 

2 

12 

22 

32 

42 

52 

62 

72 

82 

3 

13 

23 

33 

43 

53 

63 

73 

83 

4 

14 

24 

34 

44 

54 

64 

74 

84 

5 

15 

25 

35 

45 

55 

65 

75 

85 

6 

16 

26 

36 

46 

56 

GG 

76 

86 

7 

17 

27 

37 

47 

57 

67 

77 

87 

8 

18 

28 

38 

48 

58 

68 

78 

88 

9 

19 

29 

39 

49 

59 

69 

79 

89 

90 
91 

92 
93 
94 
95 

96 
97 
98 
99 


for  himself. 


I"!;' 


expressing  the 
numbers  he  has 
first  learned ;  he 
mustbetold,also, 
how  to  write  one 
ten  and  no  units ; 
he  will  then  be 
able  to  construct 
the  entire  table 
According  to  the  plan  suggested,  he  con- 
structs the  upper  horizontal  row — the  tens — first :  One 
ten  and  no  units,  two  tens  and  no  units,  three  tens  and 
no  units,  etc. ;  then  the  second  column,  the  numbers 
from  10  to  20;  then  the  third  column,  two  tens  and  1, 
2,  3,  ...  9  units,  etc.  He  thus  names  and  expresses 
the  numbers  from  1  to  99  inclusive.  He  can,  of  course, 
construct  with  equal  ease  the  horizontal  rows :  one  ten 
and  1,  two  tens  and  1,  three  tens  and  1  .  .  . ;  one  ten 
and  2,  two  tens  and  2,  three  tens  and  2,  .  .  .  nine  tens 
and  2 ;  but  the  emphasis  should  be  put  on  the  consecu- 
tive numbers,  counting  from  1  to  100.  Handling  his 
counters,  the  child  in  a  very  short  time  will  have  work- 


ON  PRIMARY  NUMBER  TEACHING. 


189 


ing  notions  of  numbers  from  1  to  100,  and  will  be  able 
to  interpret  the  symbol  by  selecting  the  right  number 
of  counters,  or  express  any  given  number  of  countei's 
by  the  right  symbol.  In  a  similar  way  the  child  will 
construct  the  200  table,  the  300  table,  etc. 


■i   fi 


:  i 


I.v   • 


CHAPTER  X. 


NOTATION,    ADDITION,    SUBTRACTION. 


^«i 


w? 


1 


■  "I  ^ 


JVumeration  and  Notation. — When  tlie  pupil  is 
rightly  drilled  in  the  constructive  processes  already  de- 
scribed, he  has  learned  what  a  unit  is — a  quantity  used 
to  measure  another  quantity  of  the  same  kind — and  he 
has  acquired  a  fair  idea  of  number  as  denoting  liow 
many  units  make  a  given  quantity.  If  the  naming 
and  the  notation  of  numbers  have  gone  on,  step  by 
step,  as  suggested,  with  the  development  of  these  ideas, 
all  numeration  and  notation  are  potentially  in  his  pos- 
session. He  has  learned  to  count,  and  to  express  his 
counts  in  symbols.  He  has  learned  the  names  of  the 
numbers  from  one  to  ten  ;  he  has  learned  to  use  the 
10-unit  quantity — the  hundred  standard  units — as  a  new 
unit,  counting — as  with  the  ones^  the  units  of  reference 
— one^  two^  .  .  .  ten  /  he  has  learned  to  use  this  10-unit 
— a  thousand  of  the  standard  units — as  a  new  unit  of 
measure,  and  to  count  by  thousands,  one,  two,  etc.  He 
has  learned  also  the  names  of  the  numbers  between  ten 
and  twenty,  twenty  and  thirty,  thirty  and  forty,  etc. ; 
between  one  hundred  and  two  hundred,  two  hundred 
and  three  hundred,  etc. ;  also,  the  names  of  the  num- 
bers between  one  thousand  and  two  thousand,  two  thou- 
sand and  three  thousand,  etc.     In  all  this  he  has  done  a 

190 


"    SI 


NOTATION,  ADDITION,  SUBTRACTION. 


191 


great  deal  for  liimself.  With  here  and  tlierc  an  apt  sug- 
gestion, he  has  been  able  to  nanne  the  numbers  from  ten 
to  twenty  ;  to  name  the  tens  (three  ty  =  thirty,  etc.)  up 
to  ten  tens  (the  child  will  probably  call  it  ten-ty),  when 
he  is  given  a  new  name  for  the  new  unit  of  measure — 
one  Imndred  ;  and  so  on  with  the  other  numbers  that 
he  has  been  using. 

The  Syinhols. — From  idea  to  name,  and  name  to 
symbol,  is  the  order.  As  he  names  with  but  little  aid 
from  suggestion,  so  he  needs  but  little  assistance  in  nota- 
tion, lie  is  given  the  digits  and  the  zero,  and  knows 
their  significance.  He  knows  that  1  denotes  any  one 
unit  of  measure  w^hatever,  and  that  0  denotes  no  unit,  no 
quantity,  lie  is  told  that  the  expression  for  a  10-unit 
quantity  is  10,  meaning  one  10-unit  and  no  one-Mi\\t.  The 
unit  of  reference,  the  one-wmt,  being  called  simply  the 
unit,  he  will  pass  immediately  to  expressions  for  one  ten 
and  one  unit,  one  ten  and  two  units,  one  ten  and  three 
units, .  .  .  one  ten  and  ten  units — i.  e.,  two  tens,  or  twenty. 
He  will  also  express  without  any  help  two  tens  and  no 
units,  three  tens  and  no  units,  etc. ;  then  two  tens  and 
one  unit,  two  tens  and  two  units,  up  to  ten  tens  and  no 
units,  which  he  will  write  at  once  as  100 ;  counting  up 
and  expressing,  that  is,  one  ten  (10),  two  tens  (20),  three 
tens  (30),  .  .  .  ten  tens  (100),  just  as  he  has  counted 
up  and  expressed  the  one- units,  1,  2,  3,  4  .  .  .  10.  But 
he  knows  that  the  ten  tens  make  a  new  unit  of  measure, 
viz.,  07ie  hundred  ;  he  sees  the  significance  of  the  1 
here,  as  in  1  (one-unit)  and  in  10,  and  counts  and  ex- 
presses his  counts  in  exactly  the  same  way — one  hun- 
dred (100),  two  hundred  (200),  three  hundred  (300), 

etc.,  to  ten  hundred  (1000).     He  has  learned  also  that 
14 


1 ,  \  (Bf*  \ 


,.|VM1 


|i* 


i;; 

'     nil 

k  i: « 


"I  I'  f 
If; 


n 


^■;il 


Im'V 


,"1  ^' 


if 


192 


THE  PSYCHOLOGY   OF  NUMBER. 


ten  of  tlie  hundred  units  make  a  new  unit  of  measure — 
the  o7?<?-thousand  unit — and  he  now  sees  the  signilicance 
of  the  1  in  tliis  place  {the /mirth  place),  and  can  go  on 
counting  and  expreu^sing  his  counts  of  the  ten-thousand 
units.  Proceeding  thus  to  any  desired  extent,  he  has 
almost,  unaided,  mastered  the  principles  of  the  decimal 
system — the  use  of  the  zero,  the  absolutely  unchanging 
values  of  the  digits  as  numbers,  the  values  of  the  units 
of  measure  denoted  by  any  digit  according  to  its  place  in 
the  series,  the  single  figure  denoting  so  many  one-units 
(so  many  units  of  reference — yard,  dollar,  pound,  etc.), 
the  second  figure  to  the  left  so  many  ten-units,  the  third 
60  many  hundred-units,  the  fourth  so  many  thousand- 
units,  the  fifth  so  many  ten-thousand  units,  etc.  He  will 
see  that  7,  75,  754,  always  denote  seven,  seventy-five, 
seven  hundred  and  fifty-four  respectively,  the  position 
of  the  figure  or  figures  in  each  case  giving  the  iimtj 
and  will  note  that,  in  reading  the  numbers  expressed  by 
the  figures,  the  figures  are  always  taken  in  groups  of 
three — for  example,  in  the  number  745,745,745,  each 
745  is  read  seven  hundred  and  forty-five,  the  dift'erence 
being  in  the  unit  only :  745  of  the  niUlion-Mmi,  745  of 
the  thousand-umi,  and  745  of  the  one-ymit,  the  primary 
unit  of  reference. 

The  Decimal  Point. — Since  the  pupil  knows,  if 
rightly  taught  upon  the  idea  of  measurement,  that  the 
unit  of  reference — the  metre,  the  yard,  the  dollar,  etc. — 
may  itself  be  measured  off  in  ten  parts,  or  a  hundred 
parts,  etc.,  he  will  be  curious  to  learn  how  the  parts  may 
be  expressed.  Knowing  that  1  denotes  1  unit,  or  1  ten- 
unit,  or  1  hundred-unit,  etc.,  according  to  its  position, 
he  will  be  eager  to  learn  how  it  may  be  used  to  denote 


in 


%  i 


NOTATION,  ADDITION,  SUIiTRACTION. 


193 


the  one-tenth  unit,  ten  of  which  iiuikc  up  the  one  unit 
(of  reference) ;  the  one-hundredtli  unit,  one  hundred  of 
whicli  make  up  the  unit;  the  one-thuutiandth  unit,  one 
thousand  of  which  make  up  the  unit,  etc.  lie  actually 
sees,  for  example,  that  the  metre  is  divided  into  ten  equal 
parts  (tenths),  each  of  these  into  ten  parts  (hundredths), 
etc.  llow  are  these  to  be  expressed  ?  lie  will  have 
but  little  difficulty  with  the  problem.  In  the  quantity 
expressed  by  111  metres  (or  dollars),  he  knows  that  the 
1  on  the  right  denotes  one-metre,  the  next  1  one  10- 
metre  unit,  the  third  1  one  100-metre  unit ;  and  passing 
from  left  to  right,  he  knows  that  the  second  1  denotes 
one  tenth  of  the  first,  and  the  third  one  tenth  of  the 
second.  Can  we  place  another  1  to  the  right  of  the 
third,  to  denote  one  tenth  of  a  metre,  then  another  to 
denote  1-hundredth  of  the  metre,  etc.  ?  Yes,  if  we  in 
some  way  mark  off  the  figures  representing  the  sub- 
divisions of  the  unit  (metre)  from  the  multiples  of  the 
metre.  We  might  distinguish  the  I's  denoting  metres 
from  those  denoting  parts  of  the  metre  by  drawing  a 
verticalline  between  them — thus:  111  |  111 — the  figures 
to  the  left  of  the  line  denoting,  respectively,  1  metre, 
1  ten-metre,  1  hundred-metre ;  and  those  to  the  right 
denoting  1-tenth  metre  (decimetre),  1-hundredth  metre 
(centimetre),  and  1-thousandth  metre  ;  and  both  consti- 
tuting one  series  governed  by  the  same  law,  namely, 
increasing  throughout  from  right  to  left  by  using  ten 
as  a  multiplier^  and  decreasing  throughout  from  left  to 
right  by  using  ten  as  a  divisor — i.  e.,  one  tenth  as  a 
multiplier.  But,  instead  of  such  a  separating  line,  it  is 
more  convenient  to  use  a  dot,  called  the  decimal  point, 
to  mark  the  place  of  the  figure  expressing  the  single 


^iir  « 


J I 


m-> 


V,:. 


|.|; 


194 


THE  PSYCHOLOGY  OF  NUMBER. 


iinil  — i.  e.,  the  unit  of  reference.  Thus  the  number  ex- 
pressed by  the  ones  previously  given  will  be  expressed 
by  lll'lll.  The  iirst  figure  to  the  left  of  the  unit-figure 
Avhose  position  is  thus  marked,  for  example,  in  453»453 
metres,  denotes  tens/  the  first  figure  to  the  right, 
tenths  I  the  second  figure  to  the  left,  himdreds  j  the 
second  figure  to  the  right,  hundredths  ^  the  third  figure 
to  the  left,  thousands  I  the  third  figure  to  the  right, 
thousandths,  etc. 

It  will  be  readily  observed,  too,  that  the  figures  to 
the  right  of  the  decimal  point  are  read  in  groujis  of 
tliree,  just  as  those  to  the  left  are.  As  denoting  a  mim- 
her,  453  is  always  four  hundred  and  fifty-three.  In  this 
example  it  is  on  the  left  side  of  the  decimal  point,  453 
inetres  (primary  units) ;  on  the  other  it  is  453  milli- 
tnetres,  etc.  Here,  as  everywhere,  there  must  be  a  good 
deal  of  drill,  in  order  that  the  pupil  may  acquire  perfect 
facility  in  reading  and  writing  numbers ;  this  means, 
ability  to  read  automatically  any  number  and  its  unit 
of  measure,  and  similarly  to  express  any  quantity  that 
may  be  named.  For  example  :  naming  each  period  ac- 
cording to  its  unit  of  measure,  name  the  first  period 
(group  of  three  figures)  to  the  left — the  units  period  ; 
the  second — the  thousands  (thousand-unit)  period  ;  the 
third  to  the  left — the  millions  period  ;  the  first  period 
to  the  right — the  thousandths  period  ;  the  second  to  the 
right — the  millionths  period.  Make  the  figure  7  ex- 
press billionths,  hundred  thousands,  tens,  tenths,  bill- 
ionths  ;  make  45  express  tens,  hundreds,  thousandths, 
millionths,  etc.  Care  is  to  be  taken  to  name  correctly 
the  measuring  units  in  the  periods  to  the  right — for 
example,  '00573  is  five  hundred  and  seventy-three  hun- 


NOTATION,  ADDITION,  SUBTRACTION. 


195 


dred-thousandths I  •0006734  is  six  thousand  seven  hun- 
dred and  thirty-four  ten-millionths. 


ADDITION    AND    SUBTRACTION. 

Addition. — In  addition,  as  we  have  seen,  we  work 
from  and  within  a  vague  whole  of  quantity  for  the  pur- 
pose of  making  it  definite.  If  a  quantity  is  measured 
by  the  parts — 2  feet,  3  feet,  4  feet,  5  feet — we  do  not 
arrive  at  the  definite  measurement  by  simply  counting 
the  nuirJjer  of  the  parts ;  we  have  to  count  the  number 
of  the  common  unit  of  measure  in  all  the  parts,  and  so 
find  the  whole  quantity  as  so  many  times  this  common 
unit.  In  learning  addition,  the  countings  are  associated 
with  intuitions  of  groups  of  measuring  units,  and  the 
results  stored  up  for  practical  use.  The  pupil  who  has 
been  properly  trained  does  not,  in  the  foregoing  ex- 
ample, start  with  2,  count  in  the  3  by  ones,  then  the  4 
by  ones,  etc.,  though  this  counting  was  part  of  the  ini- 
tial stage  even  when  aided  by  the  best  arrangements  of 
objects,  by  which  he  at  last  perceives  that  5  +  4  =  9, 
without  now  counting  by  ones.  Addition  may  there- 
fore be  considered  as  the  operation  of  finding  the  quan- 
tity which,  as  a  whole,  is  made  up  of  two  or  more  given 
quantities  as  its  parts.  The  parts  are  the  addends  (quan- 
tities to  be  added),  and  the  result  which  explicitly  de- 
fines the  quantity  is  the  smn.  It  follows  that  in  every 
addition,  integral  or  fractional,  all  the  addends  and  the 
sum  must  be  quantities  of  the  same  kind — i.  e.,  each  and 
all  must  have  the  same  measuring  unit.  Kot  only  is  it 
impossible  to  add  5  feet  to  4  minutes ;  it  is  impossible  to 
add  5  feet  to  4  rods — i.  e.,  to  express  the  whole  quantity 
by  a  mimher  (denoting  so  many  units  of  measurement) — 


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196 


THE  PSYCHOLOGY  OF  NUMBER. 


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without  first  expressing  the  addends  in  the  same  unit  of 
measurement. 

Thorough  mastery  of  the  addition  tables  must  be 
acquired,  and  rapidity  and  accuracy  in  both  mental  and 
written  work.  Exercises  on  combinations,  not  with  the 
single  units  only,  but  with  the  10-units,  the  100-units, 
the  1000-units — any  units  of  which  the  pupil  has  ac- 
quired a  fair  working  idea — will  greatly  aid  (are  a  ne- 
cessity) in  attaining  this  knowledge  of  sums  and  diifer- 
ences  as  well  as  skill  in  its  application.  Practical  facility 
in  handling  numbers  mtist  be  acquired,  at  first  with 
partial  meaning,  afterwards  with  full  meaning  of  the 
operations.     Some  additional  points  may  be  noticed  : 

1.  Use  must  be  made  of  the  knowledge  of  the  tens 
as  acquired  in  the  way  referred  to ;  for  example,  if 
8  +  8  =  16,  then  18  -f  8  =  26,  28  -f-  8  =  36,  etc.,  should 
follow  instantly  as  a  logical  consequence.  The  pupil 
may  at  first  "  make  np  "  to  the  next  ten  by  separating, 
for  example,  8  into  2  +  6,  giving,  that  is,  18  +  2  -F  6  = 
20  +  6  =  26,  just  as  at  first  lie  may  get  the  sum  8  -f  8 
through  the  steps  8  +  2-h6  =  10-|-6  =  16..  But  in 
all  cases  the  intermediate  step  should  be  dispensed  with 
as  soon  as  possible,  and  the  perception  of  the  addends — 
for  example,  28  +  8 — should  instantly  suggest  the  sum 
36,  no  matter  what  the  kind  or  magnitude  of  the  unit 
that  may  be  used. 

2.  In  this  connection  it  may  be  noticed  that  expert- 
ness  in  two-column  addition,  summing  such  numbers 
as  75,  68,  no  matter,  again,  what  the  unit  may  be,  can 
be  easily  acquired — both  acquired  and  used  with  the 
greatest  interest.  There  is  hardly  a  more  interesting 
exercise  in  that  "mental"  practice  which  is  essential 


u 


NOTATION,  ADDITION,  SUBTRACTION.  197 

f^mri  the  heginnmg  to  the  end  of  the  entire  course  in 
arithmetic,  if  knowledge,  power,  and  skill  are  to  be 
really  and  tlioronghly  gained.  Tims,  in  finding  the 
sum  of  78  and  89,  the  mental  movement  would  he  the 
sum  of  the  tens,  the  sum  of  the  units,  the  tens  and  the 
units  in  the  latter,  the  total  in  tens  and  units.  Very 
soon  the  two  "sums"  are  obtained  simultaneously,  and, 
with  a  little  practice,  the  total  (15  tens,  1  ten,  7  units) 
is  named  on  the  instant.  With  a  degree  of  facility  in 
adding  by  single  columns,  it  is  not  far  to  equal  facility 
in  adding  by  double  columns. 

3.  There  should  be  also  plenty  of  mental  practice 
in  addition  (and  subtraction)  by  equal  increments. 
Count  by  2's  from  2  to  24 ;  by  Vs  from  1  to  31 ;  by 
3's  from  3  to  36,  from  1  to  37,  etc. 

4.  It  affords  excellent  practice  in  written  work  to 
set  down  separately  the  sum 
of  each  colwrnn,  the  right- 
hand  figure  of  each  column- 
sum  being  placed  under  the 
column  from  which  it  is  de- 
rived, and  the  other  figures  in 
their  order  diagonally  down- 
ward to  the  left.  These  par- 
tial sums  are  then  added  to- 
gether to  obtain  the  total ; 
thus : 

In  this  example  the  sum  of 
the  first  column  is  42 ;  the  2 
is  placed  under  the  first  col- 
umn, and  the  4  under  the 
second    column    in  the  line  42  -|-  51  -f  45  +  57  =  195 


1 


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$9874 

28 

8768 

29 

3425 

14 

8267 

23 

1 

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16 

9341 

17 

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2345 

14 

8273 

20 

2834 

17 

6443 

17 

1 

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45 

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5454 

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195 

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198 


THE  PSYCHOLOGY  OF  NUMBER. 


below  tliat  of  the  2.  The  sum  of  the  second  cohimn  is 
61 ;  the  1  is  placed  under  the  second  column  on  the  left 
of  the  2,  and  the  5  is  placed  on  the  left  of  the  4.  The 
sum  of  the  third  column  is  45 ;  the  5  is  placed  under  the 
third  column  just  to  the  left  of  the  1,  and  the  4  diag- 
onally below  to  the  left  of  5.  The  sum  of  the  fourth 
cohinm  is  57 ;  the  7  is  placed  under  the  fourth  column 
from  which  it  was  obtained,  and  the  5  next  to  the  4  in 
the  line  below.  These  partial  sums  are  now  added  to 
get  the  total,  $62052.  Some  advantages  of  this  method 
may  be  noted  : 

(1)  It  helps  the  pupil  to  a  clearer  idea  of  the  carry- 
ing process. 

(2)  In  case  of  a  mistake  in  the  additions,  it  enables 
the  pupil  to  detect  the  error  more  easily.  What  pupil 
has  not  felt  the  drudgery  of  having  to  go  over  the 
whole  work  in  order  to  find  where  an  error  had  crept 
in  ?  By  this  arrangement  the  addition  of  any  column 
can  be  tested  independently  of  the  addition  of  the  pre- 
ceding column,  no  knowledge  of  the  "  carried "  num- 
ber being  required.  If  it  is  known,  for  example, 
that  an  error  has  occurred  in  the  addition  of  the  thou- 
sands, the  error  can  be  discovered  and  corrected  with- 
out adding  the  hundreds  to  ascertain  the  number 
carried. 

(3)  Because  the  columns  are  added  independently 
the  I'esult  may  be  tested  by  adding  the  digits  in  each 
row,  then  adding  these  sums  and  comparing  the  total 
with  the  total  obtained  from  adding  the  colunm-sums 
treated  as  separate  numbers.  These  two  totals  ought 
to  be  equal.  In  the  example,  the  sum  of  the  digits  in 
the  first  row  is  28,  in  the  second  29,  etc.,  and  the  total 


1  J- 
1  \ 


NOTATION,  ADDITION,  SUBTRACTION. 


199 


of  these  sums  is  195,  wliicli  is  the  same  as  the  total  of 
the  cohimn-siims,  42,  51,  45,  57. 

(4)  This  method  is  especially  useful  in  additions  of 
tabulated  numbers  which  are  to  be  added  both  verti- 
cally and  horizontally. 

5.  Another  excellent  practice,  for  the  more  ad- 
vanced student,  is  in  the  addition  of  two  numbers,  be- 
ginning on  the  left.  AVhen  the  common  plan  of  adding 
two  numbers  by  beginning  with  the  right-hand  digits 
is  becoming  monotonous,  the  new  method  may  be  prac- 
tised with  an  awakened  interest  because  of  its  novelty, 
and  at  the  same  time  a  broader  view  of  the  arithmetical 
operation  is  obtained.  The  only  point  to  be  attended 
to  is  whether  the  sum  of  any  pair  of  digits  we  are  work- 
ing with  has  to  be  increased  by  a  one  from  some  lower 
rank.  In  adding  a  pair  of  digits  of  any  order,  the  stu- 
dent at  the  same  time  glances  at  the  lower  orders  to 
see  if  a  one  is  coming  up  from  below  to  be  added.     In 

628 
adding  ^rc\y  while  adding  6  and  3,  we  see  at  a  glance 

that  their  sum  is  not  to  be  increased,  and  write  down  9 
at  once ;  in  adding  the  next  pair,  5  and  2,  we  instantly 
see  that  their  sum  is  to  be  increased  by  07)C  from  the 
sum  of  the  next  pair  (9  +  8),  and  we  instantly  write 
down  8.  The  student  should  practise  this  method  till 
he  can  use  it  with  ease.  He  may  exercise  himself  to  any 
extent  by  writing  down  two  numbers  and  finding  their 
sum,  then  adding  this  sum  to  the  last  of  the  two  num- 
bers, then  this  sum  to  the  preceding  sum,  etc.  As  ad- 
dition is  a  further  development  of  the  fundamental 
process  of  counting,  and  is  itself  "the  master  light  of 
all  our  seeing  "  in  numerical  operations,  perfect  facility 


m 


III 


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I-     !| 


200 


THE  PSYCHOLOGY  OF  NUMBER. 


^•1-^' 


m  \ 


y 


should  be  acquired,  though  not,  as  before  said,  by  ex- 
cluding all  other  ideas  and  operations  till  this  perfec- 
tion is  attained.  Get  complete  possession  of  addition, 
with  full  knowledge  of  numbers^  if  possible;  without 
it,  if  necessary. 

SMraction. — Addition  and  subtraction  are  inverse 
operations.  The  one  implies  the  other,  and  in  primary 
operations  the  two  should  go  together,  with  the  em- 
phasis on  addition.  Subtraction  in  actual  operations 
with  objects  would  seem  logically  to  precede  addition. 
If  we  wish  to  get  a  definite  idea  of  a  14-unit  quantity, 
and  separate  it  into  two  known  parts  of  8  units  and  6 
units  each,  it  seems  that  logically  the  6  unit-quantity 
is  taken  away  from  the  whole,  and  both  the  minor  quan- 
tities are  recognised  as  parts  of  the  whole  before  the 
final  process  of  constructing  the  whole  from  the  parts 
is  completed.  There  is  no  need,  therefore,  of  making 
a  complete  separation  between  these  two  operations. 
On  the  contrary,  they  should  be  taught  as  correlative 
operations,  wnth  addition  slightly  prominent  first  for 
reasons  already  set  forth. 

From  what  has  been  shown  as  to  the  logical  and 
psychological  relation  between  addition  and  subtrac- 
tion, it  appears  that  subtraction  is  the  operation  of 
finding  the  part  of  a  given  (juantity  which  remains 
after  a  given  part  of  the  quantity  has  been  taken  away. 
As  in  addition,  so  in  subtraction,  all  the  quantities  with 
which  we  are  working — minuend,  subtrahend,  remain- 
der— must  have  the  same  unit  of  measurement.  Further, 
as  in  addition  we  are  working  from  and  within  a  vague 
whole  by  means  of  its  given  parts,  so  in  subtraction  we 
are  working  from  a  defined  whole,  through  a  defined 


m 


NOTATION,  ADDITION,  SUBTRACTION. 


201 


it 


remain- 


part,  in  order  to  make  the  vaguely  conceived 
der  "  perfectly  definite. 

Reinainder  or  Difference. — From  the  nature  of  sub- 
traction as  related  to  addition,  there  seems  to  be  no 
strong  reason  for  the  "important  distinction"  that 
should  be  noted  between  "taking"  one  number  out  of 
another  and  finding  the  difference  between  two  num- 
bers. We  can  not  take  away  a  given  portion  of  a  given 
quantity  (to  find  the  remainder)  without  conceiving  this 
given  portion  as  part  of  the  whole ;  we  can  not  get  a 
definite  idea  of  the  "difi^erence"  between  two  measured 
quantities  without  conceiving  the  less  as  a  part  of  the 
greater.  If  $5  is  given  as  a  part  that  has  been  taken 
from  $9,  w^e  primarily  count  from  5  to  9  to  find  the 
remainder.  If  $5  and  $9  are  given  as  two  quantities, 
we  have  to  count  from  5  to  9  to  determine  the  differ- 
ence.    We  have  to  conceive  the  $5  as  a  part  of  the  $9. 

If  the  preliminary  work  of  parting  and  wholing  to 
develop  good  ideas  of  number  and  numerical  processes 
has  been  rationally  done,  there  will  be  but  little  difli- 
culty  in  the  actual  operation  in  formal  subtraction.  The 
following  points  with  respect  to  the  long-time  mystic 
operations  of  "borrowing  and  carrying"  may  be  no- 
ticed : 

1.  The  operation  involved  in,  e.  g.,  75  —  38,  may  and 
should  be  made  perfectly  clear  by  counters.  The  ten-unit 
in  its  relation  to  the  unit  has  been  made  clear  through 
many  constructive  acts.  The  mental  process  here, 
then,  is  indicated  simply  by 

60  -f  (10  -f  5) 
—  30  -     8 

=  30  -f     2  -H  5  :=  37. 


?SM 


^r 


i 


202 


THE  PSYCHOLOGY  OF  NUMBER. 


If  the  pupil  has  acquired  facility  in  the  addition  com- 
binations, the  operation  of  adding  10  and  5  and  taking 
8  from  the  sum  (getting  7)  is  probably  as  easy — may 
become  as  easily  automatic — as  taking  8  from  the  10 
and  adding  5  to  the  difference  (getting  7).  But  the 
meaning  and  identity  of  both  processes  can  be  made 
perfectly  clear.  The  pupil  may  lind  it  at  lirst  a  little 
easier  to  take  8  from  the  "  borrowed  "  10  and  add  5  to 
tlie  remainder  (2),  than  to  add  5  to  the  borrowed  10 
and  take  8  from  the  sum  15.  But,  in  any  case,  these 
analytic  acts  are  to  lead  to  the  clear  comprehension  of 
the  process,  and  especially  to  its  automatic  use.  There 
should  be,  of  course,  large  practice  in  finding  the  differ- 
ences of  pairs  of  tens,  as  well  as  in  finding  their  sums. 

2.  The  second  method  of  explaining  the  "  borrow- 
ing and  carrying  "  in  subtraction — that  of  adding  equal 
quantities  to  minuend  and  subtrahend — may  be  made 
equally  clear.  That  the  difference  between  two  quan- 
tities remains  the  same  when  each  has  received  equal  in- 
crements, the  pupil  will  discover  for  himself  by  "doing" 
such  operations.  In  75  —  38  we  add  one  ten-unit — i.  e., 
ten  ones — to  the  5  ones,  and  subtract  8,  as  in  the  first 
case  considered  ;  i.  e.,  15  —  8,  or  10  —  8  -f  5  ;  we  then 
increase  by  1-ten  the  3  tens  in  the  subtrahend,  getting  4 
tens,  which  we  take  from  the  7  tens.  This  process  is  not 
a  direct  solution  of  the  problem,  but  it  is  one  that  can 
be  made  quite  intelligible.  There  appears  to  be  but  little 
difference  in  psychological  complexity  between  the  two 
methods.  In  both  methods  8  is  to  be  taken  from  15 — 
i.  e.,  we  have  10  -f  5  —  8.  In  the  method  of  borrowing 
from  the  tens,  we  have  to  bear  in  mind,  when  we  come 
to  the  subtraction  of  the  tens,  that  the  actual  number 


NOTATION,  ADDITION,  SUBTRACTION. 


203 


,5> 


of  tens  to  be  dealt  with  is  one  less  tlian  the  number  of 

written  tens.    In  the  case  of  equal  additions,  we  have  to 

bear  in  mind  that  the  actual  number  of  tens  to  be  dealt 

with  is  one  more  than  the  number  of  written  tens. 

3.  Probably  the  best  way  to  treat  subtraction  is  the 

method  based  on  the  fact  that  the  sum  of  the  remainder 

and  subtrahend  is  equal  to  the  minuend.     If  we  wish, 

for  example,  to  find  the  difference  between  $15  and  $8, 

we  make  up  the  8  to  15,  i.  e.,  count  from  8  up  to  15, 

noting  the  new  count  of  7,  which  is  the  "  difference " 

between  8  and  15.      To  find  the   difi:'erence   between 

45   and  38  is  to  find  what  number  added  to  38  will 

45 
make  45  :    38.     The  8  units  of  the  subtrahend  can  not 

be  made  up  to  the  5  units  of  the  minuend  ;  we  make 
it  up,  therefore,  to  15  by  adding  7  units,  and  put  down 
7  as  a  supposed  part  of  the  remainder.  As  this  addition 
of  7  to  8  makes  15,  we  have  1  ten  to  carry  to  the  3  tens  of 
the  subtrahend,  making  it  4  tens,  which  requires  no  tens 
to  make  it  up  to  the  4  tens  of  the  minuend  ;  the  remain- 
der is  therefore  7.  Proceed  similarly  with  75  —  38,  etc. 
Take  an  example  with  larger  numbers.  From 
873478  take  564693— that  is,  find  what  number  added 
to  the  latter  will  give  a  result  equal  to  the  for-  ^^^ 
mer.  Write  the  subtrahend  under  the  minu-  — '— — 
end,  as  in  the  margin,  so  that  the  figures  of  o/io^ok 
the  same  decimal  order  shall  be  in  the  same 
column.  To  3,  the  right-hand  figure  of  the  subtrahend, 
5  must  be  added  to  make  up  8,  the  right-hand  figure 
of  the  minuend ;  this  is  the  right-hand  figure  of  the 
remainder.     AVe  add  8   to  9,  making  the  9  up  to  17 


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204 


THE  PSYCnOLOGY  OF  NUMBER. 


(ten-nnit),  and  putting  down  8  as  the  second  figure  of 
tlie  remainder.  We  carry  tlie  1  (hundred)  from  the 
17  (ten)  to  the  6  hundred  in  the  subtrahend,  making 
it  7  (hundred) ;  this  7  (hundred)  is  made  up  to  14  (hun- 
dred) by  adding  7  (hundred),  which  is  set  down  in  the 
tliird  place  of  the  remainder ;  carrying  1  from  the  14 
to  the  4  (thousand)  in  the  subtrahend,  we  have  5  (thou- 
sand), which  is  made  up  to  13  (thousand)  in  the  minu- 
end by  adding  8  (thousand),  and  8  is  set  down  in  the 
thousands'  place  in  the  remainder.  Similarly,  carry- 
ing 1  from  the  made-up  13  to  the  next  figure,  6,  of  the 
minuend,  we  have  7,  which  requires  7iothing  to  make  it 
up  to  7,  and  a  zero  is  therefore  set  down  in  the  10- thou- 
sands' place  in  the  remainder;  finally,  5  requires  3  to 
make  it  up  to  8,  and  so  3  is  set  down  as  the  last  figure 
of  the  remainder.  Using  italics  to  denote  the  numbers 
to  be  set  down  in  figures  as  the  remainder^  the  state- 
ment of  the  mental  process  will  be  :  3  and  Jlve^  eight ; 
9  and  eighty  seventeen ;  7  and  seven,  fourteen ;  5  and 
eight,  thirteen ;  7  and  naught,  seven ;  5  and  three, 
eight.  After  some  practice  the  minuend-sums  need 
not  be  pronounced,  and  we  shall  have  simply  3  and 
Jwe,  9  and  eight,  etc. 

This  method  is  usually  adopted  in  making  change, 
and  may  be  used  with  great  facility  in  making 
calculations  involving  both  additions  and  sub-  ^^^^^^ 
tractions.  Thus,  suppose  a  merchant,  having  2714 
$19128  in  bank,  cheques  out  the  sums  $2714,  ^^^^ 
$996,  $3952,  $166,  $7516,  how  much  has  he 
remaining  in  bank  ?  The  several  subtrahends 
are  arranged  in  columns  under  the  minuend, 
just  as  in  addition.     Add  the  subtrahends  and 


166 

7516 

$3784 


NOTATION,  ADDITION,  SUBTRACTION. 


205 


make  up  to  the  minuend  in  the  way  described,  setting 
down  the  making-up  number.     The  process  is — 

Ist  column:  12,  14,  20,  2-1  and  four,  28 — carry  2; 

2d        "  3,  9,  14,  28,  24  and  eifj/it,  32— carry  3 ; 

3d        "  8,  9,  18,  27,  34  and  seve^i,  41— carry  4 ; 

4th       "  11,  U,  16  and  three,  Id  ; 

this  makes  up  the  19  (thousand)  of  the  minuend,  and 
the  whole  "  making  -  up "  number,  or  remainder,  is 
$3784,  the  amount  of  money  the  merchant  has  left  in 
bank.  The  principle  of  "  carrying  "  is  exactly  that  of 
addition.  We  are  making  up,  by  successive  partial 
addends,  a  smaller  number  to  a  greater.  When  we 
have  come  to  24  (tens) — for  instance,  in  the  second 
column  in  the  example — we  add  8  (tens)  to  make  it  up 
to  32  (tens),  and  so  have  1  ten  more — i.  e.,  three  in  all — 
to  carry  to  the  next  " making-up"  column. 

There  seems  to  be  no  good  ground  for  the  assertion 
sometimes  made  that  this  method  is  illogical,  and  wastes 
a  year  or  more  of  the  pupil's  time.  The  first  statement 
is  refuted  by  the  psychology  of  number ;  the  second,  by 
actual  experience  in  the  schoolroom.  If  to  think  from 
15  down  to  7  is  logical,  it  would  be  no  easy  task  to  show 
that  to  think  from  8  up  to  15  is  illogical.  We  can  nei- 
ther think  down  in  the  one  case  nor  up  in  the  other 
without  thinking  of  a  measured  whole  of  15  units  as 
made  up  of  two  parts,  one  of  7  units,  the  other  of  8 
units.  As  a  conscious  process,  8  +  7  =  15  carries  with 
it  the  inevitable  correlates  15  —  8  =  7,  15  —  7=  8. 
From  what  has  been  shown  as  to  the  relations  of  the 
fundamental  operations,  it  might  even  be  inferred  that 
if  there  is  any  difference  in  difiiculty  between  the 
making-up  method  and  the  taking-away  method,  the 


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206 


THE   PSYCHOLOGY  OP  NUMBER. 


difference  is  in  favour  of  the  making-up  method,  as  in- 
volving less  demand  upon  conscious  attention.  How- 
ever this  may  be,  it  is  certainly  known  from  actual 
knowledge  of  school  practice  that  pupils  who  have  been 
instructed  under  psychological  methods  have  had  but 
little  diiticulty  in  comprehending  the  making-up  method, 
and  have  quickly  ac(juired  skill  in  the  application  of  it. 
Fmiilainental  Principles  of  Addition  and  Sub- 
traction.— When  a  quantity  is  expressed  by  means  of 
several  terms  connected  by  the  signs  -f  a,nd  — ,  the  ex- 
pression is  called  an  aggregate  ',  and  when  the  several 
operations  are  performed  the  result  is  the  total  or  sum 
of  the  aggregate.  Some  of  the  fundamental  principles 
connecting  the  operations  of  addition  and  subtraction 
are : 

(1)  If  equals  be  added  to  equals,  the  wholes  are 
equal. 

(2)  If  equals  be  subtracted  from  equals,  the  remain- 
ders are  equal. 

(3)  Adding  or  subtracting  zero  from  any  quantity 
leaves  the  quantity  unchanged. 

(4)  Changing  the  order  of  performing  the  additions 
and  subtractions  in  any  aggregate  does  not  change  the 
total  or  sum  of  the  aggregate. 

The  pupil  can  use  these  principles,  and  abstract  rec- 
ognition of  them  will  come  in  good  time. 


I! 


•! 


CHAPTER  XI. 


t|'1 


MULTIPLICATION    AND    DIVISION. 

Multiplication. — From  the  pieceding  discussion  (see 
especially  page  109  et  seq.)  of  multiplication  as  a  stage 
in  the  development  of  number,  it  is  clear  that  certain 
points  are  to  be  kept  steadily  in  view,  if  the  process  is 
to  be  made  really  intelligible  to  the  pupil. 

1.  It  is  not  simply  addition  of  a  special  kind.  It 
means  development  and  conscious  use  of  the  idea  of 
number — that  is,  of  the  ratio  of  the  measured  quantity 
to  the  unit  of  measure,  whatever  the  magnitude  of  the 
unit  may  be  in  terms  of  minor  units.  In  counting  with 
a  1-unit  measure,  one,  two,  three,  .  .  .  nine,  the  number 
is  known  when  the  unit  it  names  is  recognised  as  the 
ninth  in  a  series  of  nine  units  constituting  a  whole — 
when,  that  is,  the  defined  quantity  is  grasped  as  nine 
times  the  unit  of  measure. 

2.  In  the  development  of  the  measuring  process  (as 
in  the  exact  stage  of  measurement)  there  is  the  explicit 
recognition  that  the  measuring  unit  is  itself  measured 
olf  into  a  definite  number  of  minor  units.  This  gives 
rise  to  the  process  of  multiplication,  and  of  course  to  a 
more  definite  and  adequate  idea  of  numher  as  denoting 
times  of  repetition  of  the  unit  to  make  up  or  equal  the 
magnitude.  Nine  times  one  is  nine  is  understood  in 
it&  full  significance. 

15  207 


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208 


THE  PSYCHOLOGY  OF  NUMBER. 


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3.  A  quantity  expressed  in  terms  of  a  given  unit  of 
measure  is,  by  multiplication,  expressed  in  terms  of  the 
minor  units  in  the  given  unit  of  measure  ;  in  other 
words,  for  the  number  of  derived  units  in  the  quantity 
is  substituted  the  mmiher  of  primary  units  in  the  quan- 
tity. If  we  buy  7  barrels  of  flour  at  $5  a  barrel,  the 
measured  cost  is  $5x7;  seven  units  of  $5  each.  By 
multiplication  this  is  changed  to  $35 — i.  e.,  $1  x  35. 
This  product,  as  it  is  called,  this  new  measurement,  is 
not  seve?i  Jives.  It  denotes  the  same  quantity  under  a 
difierent  though  related  measurement ;  it  is  thirty-five 
ones.  In  one  of  these  measurements  the  nicmhe?'  is  seven, 
in  the  other  it  is  thirty -five. 

4.  The  multiplicand  must  always  be  regarded  as  a 
unit  of  measure — a  measure  made  up  of  primary  units ; 
and  the  operation  looked  upon  as  simply  making  the 
quantity  more  definite  by  expressing  it  in  a  better 
known  or  more  convenient  unit  of  measure. 

5.  While  the  multiplicand  as  multiplicand  must  al- 
ways be  interpreted  to  mean  measured  quantity,  we  can 
take  either  factor  as  multiplier  or  multiplicand.  This 
idea  must  be  used  from  the  first,  even  in  the  primary 
stage.  In  finding  the  number  of  primary  units  (dollars) 
in  12  yards  of  velvet  at  $5  a  yard,  there  is  no  known 
law  that  decrees  12  as  unchangeably  the  multiplier,  and 
$5  as  the  only  multiplicand.  On  the  contrary,  by  a 
necessary  law  of  mind,  every  measuring  process  has 
two  phases,  and  so  the  measurement  $5  x  12  carries 
with  it  the  measurement  $12  x  5.  Only  a  total  mis- 
conception of  number  and  the  measuring  process  could 
prompt  the  question.  How  can  12  yards  become  $12  ? 
The  proposition  $5  x  12  =  $12  x  5,  is  not  a  proposition 


ii  yi 


MULTIPLICATION  AND  DIVISION. 


209 


IS 


ion 


about  things  j  it  is  a  proposition  concerning  a  psychical 
process — the  mind's  mode  of  defining  and  interpreting 
a  certain  quantity.  This  principle  of  measurement — 
of  interchange  of  times  and  parts — is  essential  to  the 
proper  understanding  of  numerical  operations,  and  can 
from  the  beginning  be  intelligently  used.  Intelligent 
use  leads  to  perfect  mastery.  The  problem  of  multi- 
plication then  is :  Given  the  numher  of  unit-groups  in  a 
measured  quantity,  and  the  numher  of  minor  units  in 
each  unit-group,  to  determine,  from  these  related  fac- 
tors, the  number  of  minor  units  in  the  quantity. 

The  Formal  Process  of  Multiplication. — It  may 
be  well  to  consider  the  logical  steps  in  learning  the 
process  :- 

(1)  The  multiplication  of  a  quantity  by  powers  of 
ten.  Beginning  with  some  ultimate  or  primary  unit 
of  measure,  we  conceive  a  measured  quantity  as  mak- 
ing up  ten  such  units — that  is,  we  multiply  the  unit  by 
ten  ;  we  may  further  conceive  this  10  unit  quantity 
used  as  a  unit  of  measure,  and  repeated  ten  times  to 
make  up  a  larger  quantity — that  is,  the  10-unit  quantity 
is  multiplied  by  ten  to  express  this  larger  quantity  iri 
terms  of  the  minor  unit,  it  is  100  of  them,  etc.  It  has 
already  been  shown  how  the  notation  corresponds  with 
this  process.  The  1-unit  multiplied  by  10  becomes  10, 
the  10-unit  multiplied  by  10  becomes  100  ;  in  oilier 
words,  the  1  increases  10  times  with  every  removal  to 
the  left  of  the  decimal  point.  So  the  product  of  5 
ones  is  10  fives  or  5  tens — i.  e.,  50 ;  the  product  of  5 
tens  by  10  is  50  tens  or  500 — i.  e.,  6  multiplied  by 
100,  etc. 

(2)  We  may  find  the  total  product  which  measures 


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210 


THE  PSYCHOLOGY  OF  NUMBER. 


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a  quantity  by  finding  the  sum  of  partial  products.  If 
a  given  quantity  is  measured  by  4  feet  x  28,  we  may 
multiply  the  4  feet  by  20  and  by  8,  and  the  sum  of  the 
partial  products  will  be  the  total  product — in  all  28 
times  the  multiplicand.  This  is  the  basis  of  the  work 
in  long  multiplication. 

(3)  We  may  multiply  by  the  factors  of  the  multi- 
plier. This  is  using  the  relation  between  parts  and 
times.  If  we  have,  e.  g.,  a  quantity  expressed  by 
$2  X  20,  it  is  expressed  equally  by  $2  x  5  x  4 — i.  e., 
by  $10  X  4 ;  in  other  words,  we  have  made  the  meas- 
uring unit  5  times  as  large,  and  the  number  of  them 
5  times  as  small. 

In  the  following,  e.  g.,  we  take  the  multiplicand  5 
times,  getting  the  first  partial  product ;  in  multiplying 
by  4,  we  have  in  fact  taken  the  multiplicand  10  times, 
and  this  product  4  times,  obtaining  the  second  partial 
product  11,120. 


278 
45 

1390 
1112 

12510 


.  .     5  times  multiplicand. 

.  .  40  (i.  e.,  4  times  10  times  multiplicand). 

=  45  times  multiplicand. 


Special  Processes. — Special  processes  may  be  used 
in  many  cases.  These  afford  good  practice  for  mental 
work,  and  give  better  ideas  of  number  and  numerical 
operations,  as  well  as  preparation  for  subsequent  work. 
A  few  of  these  processes  may  be  noticed. 

1.  When  the  multiplier  is  any  of  the  numbers 
11  to  19,  the  product  can  be  obtained  in  one  line, 
thus : 


MULTIPLICATION  AND  DIVISION. 


211 


8765 
19 

166535 


(( 


(( 


li 


Nine  5's,  45 — carry  4 ; 

6's,  58  and  Jive,  63 — carry  6; 
T's,  69  and  six,  75 — carry  7 ; 
8's,  79  and  sevefi,  86 — carry  8 ; 
8  and  eight,  16. 

The  number  in  italics  is  in  each  case  the  number  in 
the  multiplicand  just  to  the  right  of  the  one  multiplied. 
To  multiply  by  31,  41,  91,  it  is  best  to  write  the  multi- 
plier over  the  multiplicand,  and  use  the  multiplicand  itself 
as  the  partial  product  from  the  digit  1  in  the  multiplier. 

For  example : 

the  multiplier. 

product  by  1. 
product  by  8  (tens). 

product. 


81 

96478567 

771829536 


7814773927  .  . 

The  product  can  be  obtained  in  one  line,  as  in  multiply- 
ing by  19,  but  there  is  greater  risk  of  error  in  the  mental 
working.  Such  examples  as  84  X  76  afford  interesting 
and  useful  mental  practice.  Multiplying  crosswise  and 
summing  the  products,  76  tens  ;  multiplying  the  units, 
2  tens  4  units ;  multiplying  the  two  tens,  56  hundreds ; 
hence  63  hundreds,  8  tens,  and  4  units — i.  e.,  6384. 

2.  Practice  in  finding  the  squares  of  numbers  is  very 
useful.  The  rule  for  finding  the  square  of  the  sum  of  two 
numbers  and  the  difference  between  the  squares  of  two 
numbers  may  be  readily  arrived  at.  For  example,  multi- 
ply 85  by  85 : 

80-1-5 

80+5 

80'  -j-    5  X  80 

+    5X80-f5' 

80»  +  10  X  80  +  5'  =  7225 


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212 


THE  PSYCHOLOGY  OF  NUMBER. 


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This  may  be  illustrated  by  intuitions,  symbolising  units 
by  dots.     Let  the  following  indicate  the  square  of  7 

(5  ~|-  2).  We  see  at  once  that  to 
make  up  the  whole  square  there  is 
(i)  the  square  of  5,  {it)  5  taken  twice, 
and  (iii)  the  square  of  2 — that  is, 
the  square  of  the  first  number,  twice 
the  product  of  the  first  by  the  sec- 
ond, and  the  square  of  the  second 
number.  It  will  be  readily  seen 
that  the  difference  of  the  two  squares  {V  —  5')  is  twelve 
times  two  j  but  twelve  is  the  sum  of  the  numbers  and 
two  their  difference.  Does  this  hold  for  other  num- 
bers? The  pupil  will  be  greatly  interested  in  discov- 
ering for  himself  the  general  principle :  the  difference 
of  the  squares  of  two  numbers  is  equal  to  the  sum  of 
the  numbers  multiplied  by  their  difference. 

If,  in  the  figure,  he  compares  the  square  of  3  with  the 
square  of  4,  of  4  with  that  of  5,  he  will  see  that  the  square 
of  any  of  these  numbers  is  got  from  the  square  of  the  next 
lower  by  simply  adding  the  sum,  of  the  numbers  to  the 
square  of  the  lower.  The  square  of  3  is  9  ;  the  square  of 
4  is  9  (=  3")  -f  (3  -f  4) ;  the  square  of  5  is  16  +  (4  +  5) ; 
and  the  square  of  7  is  36  +  (^  +  ^)-  The  pupil  will  de- 
duce for  himself  that,  given  the  square  of  any  number, 
the  square  of  the  next  consecutive  number  is  obtained 
by  adding  the  sum  of  the  numbers  to  the  given  square. 
All  these  principles,  and  many  others,  may  be  made 
the  basis  of  exercises  equally  interesting  and  useful  in 
mental  arithmetic :  Square  of  95  ;  multiply  95  by  105, 
(100  -  5)  (100  -f  5) ;  295  by  305,  (300  +  5)  (300  -  5) ; 
the  square  of  250 ;  the  square  of  251,  etc. 


MULTIPLICATION  AND  DIVISION. 


213 


3.  The  making-iip  metliod  in  subtraction  may  be 
conveniently  used  when  the  product  of  one  number  by 
another  has  to  be  taken  from  a  given  quantity. 

From  89713  take  8  times  8793.  Tlie  work  is  done 
as  follows : 


'^lii 


89713 

8793 
8 


Eight  3's,  24  and  nine  =  33 — carry  3 ; 

9's,  75  and  six  =  81 — carry  8  ; 
7's,  64:  and  three  =  07 — carry  6 
8's,  70  and  nineteen  =^  89. 


a 


a 


19369 

The  numbers  in  italics  indicate  the  remainder,  19369. 

4.  Advantage  may  often  be  taken  of  the  fact  that 
Bome  of  the  numbers  (tens,  etc.)  of  the  multiplier — and, 
once  more,  either  factor  may  be  made  the  multiplier — 
represent  a  multiple  of  some  of  the  others.  If,  for  in- 
stance, we  want  to  find  the  cost  of  2053  bags  of  flour, 
at  $3,287  a  bag,  w^e  may  use  the  latter  for  multiplier, 
and  write  only  three  partial  products : 

2053 


3287 

14371 

57484 
615  9 

^  6  7  4  8.2  1 1 


7  times. 

2  8  0  times. 

3  0  0  0  times. 


In  tliis  example  we  multiply  by  7,  and,  observing 
that  28  is  4  times  7,  we  multiply  the  first  line  of  the 
product  by  4,  getting  the  second  line ;  then  the  multi- 
plicand by  3,  taking  care,  of  course,  to  put  the  product 
in  the  thousands'  place. 

We  may  often  take  advantage  of  this  method  by 
breaking  the  order  of   finding    the  partial  products. 


1 

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214 


THE  PSYCHOLOGY  OF  NUMBER. 


Thus,  if  the  product  of  567392  by  218126  is  required, 
we  may  use  the  former  as  multiplier,  and  work  thus : 

218126 
567392 


15  2  6  8  8  2 
215056 

85505  3  92 


7  0  0  0  times. 
5  6  0,0  0  0  times. 
3  9  2  times. 


1  2  3,7  6  2,9  4  7,3  9  2 


"We  nutict-  iiiat  ^Q  is  8  times  7,  and  that  392  is  7 
times  56.  Le;^;i  i,  therefore,  with  7.  Multiplying  this 
pro<l'iCt  by  8,  we  }v->^n)  the  second  line  of  partial  prod- 
ucts; and,  jinFu^y,  \n  '-Jplying  this  second  line  by  7,  we 
get  the  third  line  of  partirii  products. 

Or  we  might  have  used  218126  for  multiplier,  ob- 
serving that  9  times  2  are  18,  and  7  times  18  is  126 ; 

thus; 

567392 
21812  6 

1134784 
10213056 

71 491392 
1  2  3,7  6  2,9  4  7,3  9  2 

Multipiying  first  by  2  (hundred  thousand) ;  multiply- 
ing this  product  by  9 ;  multiplying  this  second  partial 
product  by  7,  taking  care  as  to  the  proper  placing  of 
the  products,  we  have  the  complete  product. 

5.  Another  plan  that  affords  a  good  exercise  in 
mental  additions,  and  subsequently  proves  useful,  is 
the  method  of  finding  a  product  of  two  factors  in  a 
single  line.     To  multiply,  e.  g.,  487  by  563,  write  the 


MULTIPLICATION  AND  DIVISION. 


215 


multiplier,  with  the  digits  in  inverted  order,  on  the  lower 
edge  of  a  slip  of  paper,  thus,  3  6  51 .  Place  the  paper 
over  the  multiplicand  so  that  the  units  (3)  shall  be  just 
over  the  units  of  the  multiplicand.  The  artifice  con- 
sists in  moving  the  slip  of  paper  along  the  multiplicand, 
figure  by  figure,  till  the  last  digit  (5)  of  the  inverted 
multiplier  is  over  the  last  digit  of  the  multiplicand,  and 
taking  the  product  of  any  pair,  or  the  sum  of  products 
of  any  pairs,  of  numbers  that  may  be  in  column.     Thus : 

Three  T's,  21 — one  and  carry  2 ; 


Three  8's,  26 ;  six  T's,  ^^— eight  and 
carry  6 ; 

Three  4's,  18  ;  six  8's,  seven  5's ;  101 — 
one  and  carry  10 ; 

Six  4's,  34 ;  five  8's ;  U—four  and 

carry  7 ; 
Five  4's  and  7  carried — twenty-seven. 

The  numbers  in  italics,  taken  in  order, 
are  the  product,  274181. 


3  65 

487 

365 

487 

365 

487 

365 

487 

3  65 

487 


Proofs  of  Multiplication.— {V)  By  repeating  the 
operation  with  the  factors  interchanged.  (2)  The  prod- 
uct divided  by  either  factor  should  give  the  other  factor. 
(3)  By  casting  the  nines  out  of  the  multiplier  and  the 
multiplicand,  then  multiplying  these  remainders  together 
and  casting  the  nines  out  of  their  product ;  the  remainder 
thus  obtained  should  equal  the  remainder  from  casting 
the  nines  out  of  the  product  of  multiplier  and  multi- 
plicand. For  example,  test  the  following  by  casting 
out  the  nines ; 


"'■■n 


iKk'h 


216 


THE  PSYCHOLOGY  OF  NUMBER. 


ri, 

•  ■  i 


987761  X  56789  =  56,093,959,429. 

7      . .  out  of  product  of  2  and  8. 
Out  of  multiplicand  . .  2  X  ^  •  •  out  of  multiplier. 

7      . .  out  of  product. 

This  proof  is  not  a  perfectly  sure  test  of  accuracy.  It 
does  not  point  out  an  error  of  9,  or  of  a  multiple  of  9,  in  the 
product.  Thus,  if  0  has  been  written  for  9  or  9  for  0,  or  if 
a  partial  product  has  been  set  down  in  the  wrong  place,  or 
if  one  or  more  noughts  have  been  inserted  or  omitted  in 
any  of  the  products,  or  if  two  figures  have  been  inter- 
changed, or  if  one  figure  set  down  is  as  much  too  great 
as  another  is  too  small,  casting  out  the  nines  will  fail  to 
detect  the  error,  for  the  remainder  from  dividing  by  9 
will  not  be  affected.  Still  the  proof  is  interesting,  as 
throwing  light  upon  the  decimal  system  of  notation. 

The  Ifultiplication  Table. — The  sure  groundwork 
for  this  table  is,  of  course,  facile  mastery  of  the  addition 
and  subtraction  tables.  Though  scraps  of  it  given  from 
time  to  time — as  the  2's  and  3's  in  6 — are  perhaps  of 
no  great  value  as  contributing  to  the  making  and  mas- 
tering of  the  entire  table,  yet  some  complete  parts  of 
the  table — as,  for  example,  two  times,  three  times,  ten 
times^ — may  be  kept  in  view,  and  may  be  expertly  han- 
dled quite  early  in  the  course.  It  has  been  said  that  the 
table  is  a  grand  effort  of  the  special  memory  for  sym- 
bols and  their  combinations,  and  that  the  labour  can  not 
be  extenuated  in  any  way.  The  labour  is,  indeed,  heavy 
enough,  but  it  is  believed  that  it  may  be  somewhat  light- 
ened. The  table,  as  the  key  to  arithmetic,  must  be 
learned,  and  it  must  be  learned  perfectly — i.  e.,  so  that 
any  pair  of  factors  instantly  suggests  the  product ;  there 
must  be  no  halting  memory  summoning  attention  and 


.^1 


MULTIPLICATION   AND  DIVISION. 


217 


judgment  to  its  aid.  It  is  tlieiefore  worth  while  to 
''extenuate"  the  lahour  of  learning  it,  if  this  can  pos- 
sibly be  done.  To  this  end  some  suggestions  are  made 
which  are  believed  to  be  rational,  while  they  have  cer- 
tainly stood  the  test  of  experience. 

1.  The  Meaning  of  the  Tahle. — Pnpils  rightly  taught 
know  how  to  construct  the  table ;  they  know  what  it 
means.  The  symbol  memory,  like  every  other  kind  of 
memory,  is  always  aided  where  intelligence  is  at  work. 
In  former  times,  not  so  long  past,  the  table  used  to  be 
said  or  sung — rattled  off  in  some  familiar  tune — with- 
out a  glimmer  of  what  it  all  meant ;  but  under  rational 
instruction  the  children  know  several  important  things 
about  it,  and  the  teacher  should  use  these  things  in  less- 
ening the  labour  of  complete  mastery. 

2.  Memory  aided  hy  Intelligence. — (1)  The  pupils 
have  learned  how  to  construct  any  part  of  the  table, 
two  times,  three  times,  etc. 

(2)  They  know  exactly  what  such  construction  means, 
for  they  have  acquired  a  fair  idea  of  times — of  number 
as  denoting  repetition  of  a  measuring  unit.  They  know, 
therefore,  the  meaning  of  every  product :  6  oranges  at 
5  cents  apiece,  6  yards  of  calico  at  9  cents  a  yard,  etc. 

(3)  They  can  derive  the  product  of  any  pair  of  fac- 
tors from  the  product  of  the  immediately  preceding 
pair.  Knowing  that  6  yards  of  cloth  at  8  cents  a  yard 
cost  48  cents,  they  know  that  7  yards  cost  8  cents  more. 
Similarly  they  quickly  learn  that  if  10  oranges  cost  50 
cents,  9  oranges  will  cost  five  cents  less,  and  8  oranges 
one  ten  less,  etc.  Thus  they  will  have  various  ways  of 
constructing,  and  recovering  when  momentarily  forgot- 
ten, the  product  of  any  pair  of  digits. 


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218 


THE   PSYCHOLOGY  OF  NUMBER. 


3.  The  Commvtatwn  of  Factors. — In  learning  tlie 
table  the  relation  of  the  factors  must  be  kept  in  view. 
This  greatly  reduces  the  labour.  There  ought  to  be 
little  difficulty  in  this  if  a  fair  idea  of  the  relation  be- 
tween parts  and  times  has  been  brought  out.  At  3  cents 
apiece,  5  oranges  cost  3  cents  X  5  ;  this  is  seen  to  be  iden- 
tical with  5  cents  X  3.  Each  of  these  expresses  meas- 
ured quantity,  a  sum  of  money  ;  the  thought  "  oranges  " 
disappears  from  this  conception.  The  table  is  often 
taught  without  reference  to  this  principle,  and  so  the 
labour  of  learning  it  is  at  least  one  half  greater  than  it 
ought  to  be.  In  our  boyhood  we  learned  9  X  6  =  54, 
without  a  suspicion  that  6  X  0  =  54.  Let  us  see  to  it 
that  the  present  things  be  made  better  than  the  former. 

4.  Memory  fiirther  aided. — Associations.  In  this 
connection  the  following  suggestions  are  worthy  of  at- 
tention : 

(1)  The  thing  to  be  kept  in  view  is  that,  so  far  as  possi- 
ble, associations  are  to  he  formed  directly  Jjetween  a  jpro- 
duct  and  one  or  hoth  of  the  factors  ivhich  produce  it. 

(2)  Ten  times  is  already  learned  in  addition — in  the 
counting  of  the  tens.  The  pupil  know^s  how  to  "mul- 
tiply" any  number  by  ten  by  simply  affixing  a  zero  to 
the  number.  The  association  of  product  and  factors  is 
direct ;  the  product  is  the  multiplicand  with  the  zero 
of  the  10  affixed.     Ten  times,  then,  is  well  in  hand. 

(3)  Eleven  times  is  almost  equally  easy.  The  prod- 
uct in  each  case,  for  the  first  nine  digits,  is  directly 
associated  with  the  digit  j  the  digit  is  simply  repeated — 
11,  22,  33,  etc.  Eleven  tens  is  known  from  ten  elevens, 
and  the  other  two  products  (11  X  11,  12  X  11)  must  be 
built  up  from  this. 


MULTIPLICATION  AND   DIVISION. 


219 


ens, 
it  be 


(4)  Nine  times  can  be  formed  and  remembered  in 
a  similar  way.  The  pupil  will  note  :  («)  That  a  product 
is  made  up  of  tens  and  units.  {]))  That  in  9  times  (up 
to  10  X  9)  the  number  of  tens  is  always  one  less  than 
the  number  multiplied,  [c]  That  in  every  product  the 
sum  of  the  digits  is  9 ;  and  thus,  having  written  down 
the  tens  directly  from  the  multiplicand,  he  can  at  once 
write  the  units.  He  should  be  led  to  notice  also  that 
(i)  holds  good  as  to  the  law  of  tens  up  to  10  X  9,  after 
which  the  number  of  tens  is  two  less  than  the  multi- 
plicand up  to  and  including  20  X  9,  after  which  the 
number  of  tens  is  three  less,  etc.  He  should  note,  too, 
in  his  formed  table,  how  the  t^ns  increase  by  one  and 
the  units  decrease  by  one.  This  may  seem  somewhat 
complex,  but  it  works  well.  We  have  known  a  boy  of 
six  years  to  construct  and  learn  9  times  up  to  9  times 
10  in  fifteen  minutes. 

(5)  Probably  two  times  has  been  completely  learned 
before  a  formal  attack  is  made  upon  the  table  as  a 
whole.  There  has  been  much  practice  in  counting  by 
2's — backward  and  forw^ard — and  by  3's,  etc.  There 
seems  to  be  no  way  of  making  a  mnemonic  association 
between  a  product  and  its  factors  ;  but  addition  by  two 
is  an  easy  operation,  and  two  times  is  quickly  learned. 

(6)  In  twelve  times  (assuming  two  times)  the  memory 
can  be  aided  by  association.  The  product  of  any  multi- 
plicand may  be  obtained  by  taking  it,  the  multiplicand, 
as  so  many  tens^  and  doubling  it  for  the  units  ;  tw^elve 
times  3  =  three  tens  and  six  (twice  3)  units.  For  5  and 
up  to  9,  doubling  the  unit  gives  more  than  10,  but 
the  additions  are  easy.  12  times  5  =^five  tens  and  ten 
units   (twice  five)  -   60.      Or,   consider   the   products 


I'    *l 


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220 


THE   PSYCHOLOGY  OF  NUMBER. 


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tliiis :  tlio  products  of  1  —  4  are  12,  24,  36,  48  ;  those 
of  5  —  9  are  each  07ie  ten  more  tliari  the  multiplicand, 
and  the  units  increase  by  2's — i.  e.,  0,  2,  4,  6,  8 ;  the 
products  of  5,  6,  7,  8,  9  are  therefore  60,  72,  84,  96, 
108. 

(7)  Some  assistance  from  association  may  be  had  in 
learning  5  times  by  observing :  (a)  that  the  units  are 
alternately  5  and  0,  5  for  the  odd  multiplicands,  0  for 
the  even  ones  ;  {h)  if  the  multiplicand  is  even,  the  tens 
are  half  of  it ;  if  odd,  the  tens  are  half  the  next  lower 
number :  8X5  =  4  tens  and  0  units  ;  9X5  =  4  tens 
and  5  units,  etc.  More  advanced  students  will  take 
pleasure  in  extending  the  multiplication  table  according 
to  these  laws,  as  well  as  in  accounting  for  the  laws.  For 
example :  in  9  times,  why  are  the  tens  one  less  than  th 
multiplicand  up  to  10  X  9,  then  two  tens  less  up 
20  X  9  ?  etc.  In  8  times  why  are  the  tens  one  less 
than  the  multiplicand  up  to  5  X  8,  two  less  from  6X8 
to  10  X  8  ?  etc. 

Division. — Division  is,  w^e  have  seen,  the  operation 
of  finding  either  of  two  factors  when  their  product 
and  the  other  factor  are  given.  After  what  has  been 
said  in  Chapter  Y  upon  the  nature  of  division  and  its 
relation  to  multiplication  and  fractions,  little  further 
need  be  added,  especially  as  most  of  the  text-books  ex- 
plain clearly  enough  the  actual  arithmetical  work.  A 
few  points,  however,  may  be  briefly  noticed  :  (1)  If, 
in  the  method  of  teaching,  the  idea  of  number  as  meas- 
urement has  been  kept  steadily  in  view,  the  nature  of 
division  as  the  inverse  of  multiplication  will  be  fully 
understood.  (2)  Knowing  the  relation  of  the  factors 
in  multiplication,  the  pupil  wall,  with  but  little  difficulty, 


^ 


less 


MULTIPLICATION   AND   DIVISION. 


221 


cornpreliend  the  o[)enition  and  be  al)le  to  interpret  tlie 
results  in  every  case.  Practised  from  tlie  lirst  in  usini^ 
the  idea  of  correlation — of  number  detininc:  the  measnr- 
ing  unit  and  numher  defining  the  measured  whole — in 
both  multiplication  and  division,  he  can  tell  on  the  in- 
stant which  of  these  factors  is  demanded  in  any  problem. 
(3)  There  does  not  seem  to  be  any  necessity  for  begin- 
ning formal  division  by  the  "long  division"  process. 
The  pupil  knows  that  2  x  5  =  10,  and  that  10  -r-  r>  =  2, 
whatever  7nay  he  the  unit  of  measure.  He  knows  that 
ten  ones  divided  by  5  is  two  ones,  tliat  ten  tens  divided 
by  5  is  two  tens,  ten  hundred-units  divided  by  5  is  two 
hundred-units,  etc.  He  has  learned  that  12  units  of 
any  order  in  the  decimal  system  when  divided  by  5 
gives  2  units  of  that  order,  with  2  units  of  that  order, 
or  20  units  of  the  next  lower  order,  remaining ;  which  20 
units  on  division  by  5  gi\  's  4  units  of  that  order,  mak- 
ing the  total  quotient  24.  In  short,  if  the  pupil  has 
been  taught  to  divide  a  number  of  any  two  digits  by  any 
of  the  single  digits,  he  can  divide  any  number  by  a 
single  digit.  Thus,  suppose  497C  is  to  be  divided  by  8 : 
,  ^  here  eight  will  not  divide  4  giving  a  quotient 
~— -  of  the  same  order — i.  e.,  in  the  thousand  units  ; 
the  4  is  changred  to  40  units  of  the  next  lower 
order,  making,  with  the  9  of  that  order,  49.  This  di- 
vided by  8  gives  6,  with  1  over.  Similarly  this  1  is  10 
of  the  next  order,  which,  with  the  Y  of  that  order, 
makes  17 ;  this  divided  by  8  gives  2,  with  1  over ;  this 
1  is  10  of  the  next  order,  and  with  the  6  makes  16  of 
that  order,  which,  divided  by  8,  gives  2,  the  last  figure 
of  the  quotient.  ISTo  matter  what  the  series  of  figures, 
the  process  is  the  same,  and  the  pupil  should  experience 


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222 


THE  PSYCHOLOGY  OP  NUMBER. 


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no  real  diflSculty  if  rational  method  and  practice  have 
been  followed.     A  few  practical  points  may  be  noted  : 

(1)  The  division  by  any  power  of  10  is  as  easy  as 
multiplication  by  any  power  of  10 — is,  in  fact,  derived 
directly  from  it. 

(2)  So  with  division  by  factors  of  the  divisor,  which 
is  directly  connected  with  mnltiplication  by  factors  of 
the  multiplier.  To  the  pupil  it  will  prove  an  interest- 
ing exercise  to  discover  the  "  true  remainder."  Take, 
for  example,  5795  —  48. 

8)5795 
6)72-1 ...  3  rem.  in  ones,  the  quotient  being  724  eights. 
120  ...  4  rem.  in  8-unit  groups ; 

hence  remainder  in  07ies  is  8  X  4  +  3  =  35.  This  is 
the  old  rule  :  Multiply  the  first  divisor  by  the  second 
remainder  and  add  the  product  to  the  first  remainder. 
The  same  method  is  applicable  to  the  case  of  three  or 
more  factorial  divisors ;  apply  the  rule  to  the  last  twc 
divisions,  and  use  the  result  with  the  first  divisor  and 
first  remainder.  Or,  reduce  each  remainder  to  units  as 
it  occurs ;  for  example,  divide  2231  by  90  (=  3  X  5  X  6). 

3)2231 
5)743  unit-groups  of  3  with  rem.  2  units ; 
6)148  unit-groups  of  15  with  7'e7n.  3  groups  of  3=9  units ; 
24  groups  of  90  with  rem.  4  groups  of  15  =  60  units. 

The  remainder  is  therefore  60  +  9  +  2  =  71.  Other- 
wise, applying  the  rule  with  the  last  two  divisions : 
5  X  4 -|-  3  =  23 ;  use  this  as  th^  "second  remainder" 
from  the  "  first  divisor,"  and  remainder  23  X  3  +  2  =  71. 

(3)  In  long  division  the  multiplications  and  subtrac- 


MULTIPLICATION  AND  DIVISION.  223 

tions  may  he  combined,  as  described  under  multiplica- 
tion and  subtraction— e.  g.,  635040  -;-  864. 

864)635040(735 
3024 
4320 

(1)  Seven  4'8,  28  and  two  =  30— carry  3.  (2)  Seven 
6's,  45  and  zero  =  45— carry  4.  (3)  Seven  8's,  60  and 
three  =  63.  Tliis  gives  302,  which,  with  4  brouglit 
down,  makes  the  first  remainder.  Proceed  similarly 
with  3  and  5,  the  other  figures  in  the  quotient.  The 
student  may  note  the  application  of  the  method  in  a 
longer  operation  ;  Divide  217,449,898,579  by  56437. 
The  following  is  the  work  : 

3852967 

56437)217449898579 
481388 
298929 
167448 
545745 
378127 
395059 

Three  7's,  21  and  eight  =  29— carry  2.  Three  3's,  11 
and  three  -  14— carry  1.  Three  4'8,  13  and  one  =  14 
—carry  1.  Three  6's,  19  and  eight  =  27— carry  2. 
Tliree  5's,  17  and  four  =  21.  This  gives  48138, 
which,  with  the  8  (heavy-faced  type)  brought  down, 
makes  the  complete  first  remainder.  With  this  pro- 
ceed exactly  as  before,  and  so  on  with  the  other  re- 
mainders. 

(4)  Casting  out  the  Mnes.-^-lt  is  seen  that  9  (and 
16 


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THE   PSYCHOLOGY  OF  NUMBER. 


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of  course  3)  is  a  measure  of  9,  99,  999,  9999,  etc.— that 
is,  of  10  —  1,  100  -  1,  1000  —  1,  etc.  Hence,  if  from 
any  number  there  be  taken  all  the  ones,  and  1  from 
every  10,  1  from  every  100,  etc.,  the  remainders  from 
the  tens,  the  hundreds,  the  thousands,  etc.,  constitute  a 
number  which  is  a  multiple  of  9.  The  original  number 
will  therefore  be  a  multiple  of  9,  if  the  total  of  the  de- 
ductions made  is  a  multiple  of  9 ;  this  total  is  the  num- 
ber of  ones  -\-  the  number  of  tens  +  the  riiimher  of 
hundreds,  etc. — that  is,  this  total  is  the  sum  of  the 
digits  of  the  given  number.  For  example,  is  39273 
divisible  by  9  ? 

30000  =  3  times  10000  =  3  times  9999  and  3 

9000  =  9     "       1000  =  9     "       999    "    9 

200  =  2      "         100  =  2      "         99    "    2 

70  =  7     "  10  =  7     "  9    "    7 

1=3"  1=3"  0    "    3 

Adding  39273  =  some  multiple  of  9  and  24 

Hence  the  given  number  is  exactly  divisible  by  3,  but 
leaves  a  remainder  of  6  when  divided  bv  9,  because 
24  -T-  9  leaves  6  remainder.  The  principle  is :  any 
number  divided  by  9  leaves  the  same  remainder  as  the 
sum  of  its  digits  divided  by  9. 

To  cast  the  nines  out  of  any  number,  therefore,  is 
to  find  the  remainder  in  dividing  the  number  by  9. 
In  casting  out  the  nines  from  the  sum  of  the  digits  we 
may  conveniently  omit  the  nines  from  the  partial  sums 
as  fast  as  they  rise  above  8. 

Proofs  of  Division. — (1)  By  repeating  the  calcula- 
tion with  the  integral  part  of  the  quotient  for  divisor. 
(2)  By  multiplying  the  divisor  by  the  complete  quo- 


MULTIPLICATION  AND  DIVISION. 


225 


de- 


tient.     (3)  By  casting  out  the  nines,  as  in  multiplica- 
tion.    For  example : 

3,893,865,223  -^  179  =  21,753,437. 

4      . .  out  of  product  8x5. 
9's  out  of  divisor . .  8  X  ^  •  •  out  of  quotient. 

4      . .  out  of  dividend. 

If  there  is  a  remainder  the  method  can  still  be  applied. 
Test  the  accuracy  of 

3,893,865,378  -f- 179  =  21,753,437  j|| 
where  the  remainder  is  155. 

divisor  j"  •  •  ^X^?  2  .  .  out  of  quotient  and  remainder, 
6         .  .  out  of  dividend. 

The  disadvantages  of  this  proof  are  similar  to  those  in 
the  proof  of  multiplication  by  casting  out  the  nines. 

Fundamental  Principles  connecting  Multiplication 
and  Division. — From  the  theory  of  number  as  meas- 
urement and  numerical  operations  as  a  development 
of  the  measuring  idea,  there  are  certain  fundamental 
principles— fundamental  also  in  fractions— connecting 
the  operations  of  multiplication  and  division.  The 
principal  of  these  are  the  following  : 

(1)  If  equals  be  multiplied  by  equals,  the  products 
are  equal. 

(2)  If  equals  be  divided  by  equals,  the  quotients  are 
equal. 

(3)  If  an  expression  contains  a  series  of  multipliers 
and  divisors,  changing  the  order  of  the  multipliers  and 
divisors  does  not  change  the  value  of  the  expression. 


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226 


THE  PSYCHOLOGY  OP  NUMBER. 


The  last  principle  includes  several  principles  of  use- 
ful application,  either  implied  or  stated  explicitly  in  the 
discussions  upon  number  and  its  development. 

(a)  The  order  of  numerical  factors  may  be  changed. 
(h)  Multiplying  a  factor  by  any  number  multiplies  the 
product  by  the  same  number,  (c)  Dividing  a  factor  of 
any  number  divides  the  product  by  the  same  number. 
{d)  Multiplying  the  dividend  by  any  number  multiplies 
the  quotient  by  that  number,  (e)  Dividing  the  divi- 
dend by  any  number  divides  the  quotient  by  the  same 
number.  (/*)  Multiplying  the  divisor  by  any  number 
divides  the  quotient  by  the  same  number.  ((/)  Dividing 
the  divisor  by  any  number  multiplies  the  quotient  by 
the  same  number,  (h)  Multiplying  or  dividing  both 
divisor  and  dividend  by  the  same  number  leaves  the 
quotient  unaltered.  (^)  All  these  principles  are  neces- 
sarily involved  in  the  principles  of  number  as  already 
unfolded.  The  following  is  worthy  of  attention  :  In  an 
aggregate  whose  terms  contain  multipliers  and  divisors, 
the  inidtiplicatums  and  the  divisions  are  to  he  ^ler- 
formed  befoke  the  additions  and  the  subtractions  are 
made. 


i! 


CHAPTER  XII. 


MEASURES   AND   MULTIPLES. 

Greatest  Common  Measure. — The  pupil  who  has 
been  led  to  have  a  clear  idea  of  number — who  has  been 
taught  to  look  upon  the  unit  as  the  m£asurer — will  find 
no  difficulty  in  mastering  greatest  common  measure. 
With  all  the  preliminary  notions  he  is  familiar,  and  it 
will  be  an  easy  matter  to  pass  to  the  formal  process. 

While  in  the  illustrations  given  in  this  chapter  we 
generally  use  the  pure  number  symbols,  it  must  be  borne 
in  mind  that  here,  as  everywhere  in  number  and  nu- 
merical processes,  the  idea  of  measurement  is  to  be  kept 
prominent,  especially  in  the  introductory  lessons.  A 
common  factor  is  a  common  measure — a  unit  of  meas- 
ure that  is  contained  in  two  or  more  quantities  an  exact 
number  of  times.  A  common  multiple  is  a  definitely 
measured  quantity,  which  can  be  measured  by  two  or 
more  quantities,  themselves  measured  by  units  of  the 
same  kind  and  value  as  those  of  the  given  quantity. 
The  teacher  should  see  to  it,  then,  that  all  his  illustra- 
tions and  examples  deal  with  the  concrete  ;  that  the 
measuring  idea  be  kept  prominent  from  first  to  last. 

Easy  Resolution  into  Factors. — Taking  the  num- 
ber 15,  the  learner  sees  that  it  can  be  considered  3 
fives,  or  5  threes ;  the  five  or  the  three  is  a  measurer 

227 


n 

X 


|i'! 


m 


if  I'M?  * 


■if 


.■if 


I '  -i- .  i 


^  1  I 


\\{;. 

I 


,( 


228 


THE  PSYCHOLOGY  OP  NUMBER. 


or  measure  of  15,  and  the  equation  15  =  5  X  3  puts  in 
evidence  the  fact  that  5  and  3  are  measures  or  factors 
of  15.  Taking  35,  he  sees  the  significance  of  the  equa- 
tion 35  =  5  X  7.  He  further  notes  that  5  is  a  meas- 
ure of  each  of  the  numbers  15  and  35,  and  is  therefore 
a  common  measure.  If,  next,  the  numbers  12  and  18 
are  taken,  he  will   see  that  all   the   measures  of   12 

are — 

1,  2,  3,  4,  6,  12  ; 

and  that  all  the  measures  of  18  are — 

1,  2,  3,  6,  9,  18. 

Then  it  will  be  seen  that  1,  2,  3,  6  are  common  measures 
of  12  and  18,  and  that  while  there  are  several  such  meas- 
ures, there  is  one  that  is  the  greatest — the  one  that  will  be 
called  the  greatest  common  measure.  Before  any  pro- 
cess is  taught  the  class  should  be  exercised  in  the  work- 
ing of  easy  examples,  both  mental  and  written ;  being 
asked  to  find  common  measures,  and  the  greatest  com- 
mon measure  of  16  and  2-1,  of  24,  36,  48,  etc.  An  ad- 
ditional interest  will  be  secured  by  proposing  some  sim- 
ple practical  problems. 

It  will  be  better,  before  beginning  the  ordinary  for- 
mal treatment,  to  have  exercises  in  finding  the  greatest 
common  measure,  by  resolving  the  numbers  given  into 
their  simple  factors.  It  would  be  necessary,  then,  to  re- 
call or  develop  a  certain  fundamental  principle.     The 


division 


2)60 

3)30 

10 


is  to  be  interpreted,  first,  that  60  is  30 


twos,  and,  next,  that  the  30  twos  are  10  three-twos  or 
10  sixes  ;  and  thus  that  if  a  number  contains  the  factor 


i   ! 


MEASURES  AND  MULTIPLES. 


220 


30 


2,  and  if  the  quotient  contains  the  factor  3,  the  number 
itself  contains  tlie  factor  or  measnre  6.     Then,  since 

108  =  2  X  2  X  3  X  3  X  3, 
and    72  =  2  X  2  X  2  X  3  X  3, 
we  may  see  that  all  tlie  single  common  factors  are  2,  2, 

3,  3 ;  and  that,  therefore,  2  X  2  X  3  X  3,  or  36,  is  the 
greatest  common  measure.  Practice  on  this  method 
will  find  a  place  :  the  pupil  has  a  new  interest,  and  the 
teacher  can  take  advantage  of  it  to  secure  further  train- 
ing in  number  and  in  the  elementary  processes. 

Hie  General  Method. — But  soon  it  will  be  found 
that  this  method  is  limited,  as  its  successful  application 
depends  on  the  pupiPs  ability  to  discover  a  factor.  An 
example,  such  as.  Find  the  greatest  common  measure 
of  851  and  1073,  we  may  suppose  to  have  been  given 
the  class,  and  found  beyond  their  present  power  of 
factoring.  The  reason  for  the  failure  will  be  manifest 
to  them — their  inability  to  find  any  factor  of  either 
number.  The  need  for  some  new,  or,  it  may  be,  ex- 
tended method,  is  felt ;  and  this  need  is  the  teacher's 
opportunity  for  introducing  the  more  powerful  method, 
and  for  the  development  of  it  he  has  his  class  in  a  state 
of  healthy,  natural,  unforced  interest. 

The  Fundamental  Principles. — To  develop  the 
method,  it  would  be  well  to  turn  aside  from  the  ex- 
ample attempted  and  give  attention  to  certain  facts 
upon  which  the  method  is  based.  Taking  for  illustra- 
tion the  numbers  21  and  35,  we  see,  as  before,  that 
21  is  3  sevens  and  35  is  five  sevens.  Thus,  if  21  is 
added  to  35  we  shall  have  3  sevens,  and  5  sevens  or  8 
sevens  ;  the  seven  being  the  unit  of  measure,  or  meas- 
urer.    Similarly,  if  21  is  subtracted  from  35  the  result 


'ifi 


w 


w. 


Ill 


-h  • 


i 


m 


h«! 


I 


t'! 


la 

-^5' 


I;- 

*H   ■ 

1 

230 


THE  PSYCHOLOGY  OF  NUMBER. 


will  be  2  sevens.  Further,  if  to  21  is  added  3  times  35, 
we  liave  3  sevens  and  3  times  5  sevens — that  is,  a  cer- 
tain number  of  sevens.  This  is  seen  to  be  true  for  any 
number  of  times  seven,  any  number  of  times  eiglit, 
or  nine,  .  .  .  etc.  Actual  measurements  will  make  the 
principle  still  clearer.  Thus,  if  A  B  and  C  D  have  a 
common  measure,  it  must  measure  A  B  exactly,  and 
C  D  exactly : 


B 


C 


E 


D 


and  measuring  off  on  C  D  a  part  C  E  =  to  A  B,  the 
common  measure  must  measure  C  E  exactly,  and  there- 
fore E  D  exactly,  because  it  measures  the  whole  of 
C  D  ;  but  E  D  is  the  difference  of  the  quantities,  etc. 
In  the  same  way  E  D  may  be  measured  off  on  A  B,  and 
the  same  reasoning  will  apply.  Thus  the  pupils  are 
led  to  see  certain  general  principles,  and  to  see  them  in 
their  generality. 

1.  From  the  fact  that  if  we  take  the  sum  or  the  dif- 
ference of  21  and  35 — that  is,  of  3  sevens  and  5  sevens 
— or  the  sum  or  the  difference  of  any  number  of  times 
21  and  any  number  of  times  35,  we  are  sure  to  have  a 
number  of  sevens  (seven  representing  any  measured 
quantity  whatever),  it  is  plain  that  any  number  which 
measures  two  numbers  will  measure  their  sum  or  their 
difference,  or  the  sum  and  also  the  difference  of  any  of 
their  multiples.  The  pupils  can  be  got  to  develop  the 
general  form  of  this  principle.  If  c  is  a  common  meas- 
ure of  a  and  h,  so  that  a  =  7nCj  and  h  —  nc,  then  a-\-h 
=  mc  -\-  no,  etc. 

2.  Because  the  common  measure  of  two  numbers 
measures  their  sum,  and  because  the  minuend,  in  a  sub- 


MEASURES  AND  MULTIPLES. 


231 


traction  operation,  is  the  sum  of  the  remainder  and 
the  subtraliend,  it  is  plain  that  every  common  factor 
of  the  remainder  and  the  subtrahend  is  a  factor  of  the 
minuend. 

The  Application  of  the  Method. — We  pass  now  to 
the  application,  and  shall  take  the  numbers  851  and 
1073.  The  difficulty  has  been  that  these  numbers  are 
large,  and  in  reply  to  the  question,  What  smaller  num- 
ber will  have  in  it  any  common  factor  that  851  and 
1073  may  have  ?  there  might  be  expected  the  answer, 
1073  —  851.  But  there  must  be  an  examination  of  this 
statement. 


851 


1073 
851 

222 


If  851  and  1073  have  a  common  factor,  this  factor  will 
also  measure  222  ;  and  if  222  and  851  have  a  common 
factor,  this  factor  will  measure  1073.  Thus  the  greatest 
common  measure  of  851  and  1073  is  a  factor  of  222, 
and  the  greatest  common  measure  of  851  and  222  is  a 
factor  of  1073.  Therefore  the  greatest  common  measure 
of  851  and  222  is  the  greatest  common  measure  of  851 
and  1073.  It  will  now  be  easy  to  show  that  if  222,  or  2 
times  222,  or  3  times  222,  be  taken  from  851,  222  and 
this  remainder  will  have  for  greatest  common  factor 
the  greatest  common  factor  of  851  and  222,  and  the 
advantage  in  taking  from  851,  3  times  222  is  ap- 
parent. 

851  222 

185 


m. 


'A 


\\ 


w^ 


M 


'i 

J"-  f 


iiiii: 


111 

'•J 


.If 


M  . 


232 


THE  PSYCHOLOGY  OP  NUMBER. 


It  will  be  easy  to  follow  this  out  through  the  suc- 
cessive steps : 

185  222 

185 


185 


37 
37 


37  divides  185  exactly,  and  is  thus  the  greatest  common 
measure  of  185  and  37 ;  so  that  37  is  the  greatest  com- 
mon measure  of 

851  and  1073. 

The  class  will  now  see  that 

851  =  23  X  37 
1073  =  29  X  37 

and  a  conviction  will  be  added  to  the  proof.  Then  the 
identity  of  the  work  with  the  following  may  be  shown : 

851)1073(1 

851 

222j851(3 

185)222(1 

185 

37)185(5 
185 

We  see  now  that  a  definite  method  has  been  evolved, 
and  when  the  class  has  been  exercised  in  applying  it,  it 
may  be  well  to  explain  certain  artifices  by  means  of 
which  the  work  may  be  shortened,  or  exhibited  in  a 
neater  form.     For  example,  the  work  of  finding  the 


MEASURES  AND  MULTIPLES. 


233 


SUC- 


greatest  common  measure  of  851  and  1073,  as  given 
above,  may  be  presented  as  follows : 


1 

3 

1 

185 
185 

1073 
851 

851 

CyC)Q 

222 
185 

222 

185 

37 

37     =  G.  C.  M. 


Or  the  work  might  be  conveniently  arranged  as  in 
the  following  example :  Find  the  greatest  common 
measure  of  158938  and  531206. 


158938 

108784 

3 

2 

1 

11 

1 

5 

27 

531206 
476814 

54392 

50154 

46618 

3536 

50154 
4238 
3536 

3510 

702 

26 

702 

The  quotients  appear  in  the  middle  column,  and  the 
work  explains  itself. 

It  is  to  be  observed  that  if  any  common  factor  is 
easily  discoverable  in  the  two  given  quantities,  it  is  bet- 
ter first  to  divide  both  quantities  by  the  common  factor. 
If,  also,  a  prime  factor  is  found  in  only  one  of  the  quan- 
tities which  are  in  operation  for  the  greatest  common 
measure,  it  may  be  struck  out.  In  the  last  example, 
for  instance,  tlie  first  remainder  is  divisible  by  8,  while 
the  corresponding  number  on  the  other  side  (the  first 
divisor)  is  divisible  by  2.  We  may  therefore  divide 
this  number  by  2  and  the  other  by  8,  reserving  2  as 


I! 


^^ 


■Hi 


li'.^TT^: 


ii\ 


:'  111'''  I    ^! 


I 


1 


■If 


23i 


THE  PSYCHOLOGY  OF  NUMBEK. 


part  of  the  required  common  measure.  These  factors 
being  removed,  we  operate  with  the  quotients,  79469  and 
6799.  The  latter  divides  the  former  with  remainder 
4680 ;  this,  it  is  obvious,  lias  the  factors  40,  13,  9. 
llence,  if  the  two  original  quantities  have  a  common 
factor,  it  is  13  X  2 — a  result  obtained  by  the  actual 
work. 

This  study  of  the  measures  of  numbers  suggests 
classilications  of  numbers.  Numbers  may  be  (1)  even 
or  odd,  according  as  they  do  or  do  not  contain  2  as  a 
factor ;  (2)  composite  or  prime,  according  as  they  are 
or  are  not  resolvable  into  simpler  factors. 

Two  numbers  may  have  no  factors  in  common, 
though  each  of  them  may  be  composite ;  they  are  then 
said  to  hQ  prime  to  each  other.  It  will  be  supposed  that 
the  class  is  familiar  with  these  classifications  and  defini- 
tions before  proceeding  to  a  study  of  least  common 
multiple. 


;.?: 


lifi 


LEAST   COMMON   MULTIPLE. 

In  the  presentation  of  least  common  multiple,  it  is 
necessary — as  indeed  it  always  is  in  the  introduction  of 
a  new  process — first  to  bring  out  clearly  the  essenti-' 
facts  and  ideas  upon  which  the  process  rests.  Tlrn 
pupil  must  first  get  a  clear  idea  of  the  term  .iHj 
common  multiple,  least  common  multiple.  ..  factor 
(or  measure)  of  15  is  3 ;  15  is  called  a  mndtiple  of  l ; 
it  represents  the  quantity  that  3  exactly  measures.  The 
pupil  will  now  be  asked  to  name  different  multiples  of 
3,  say,  and  will  see  that  he  may  name  or  write  down 
as  many  as   he  chooses.      Then,  if  a  series  of  multi- 


MEASURES  AND  MULTIPLES. 


235 


pies  of  2  be  written  down,  so  that  we  have  the  two 
series : 

3,  6,  9,  12,  15,  18,  21,  21,  .  .  . 

2,  4,  (>,  8,  10,  12,  14,  16,  18,  20,  22,  24,  .  .  . 

he  will  see  that  there  are  numbers  which  are  at  the 
same  time  multiples  of  2  and  3,  and  are  therefore  cnm- 
ni07i  multiples  of  2  and  3.  These  common  multiples 
are,  here,  6,  12,  18,  .  .  .  Then,  because  we  have  started 
with  the  smallest  multiples  of  the  numbers,  6  is  the  small- 
est or  least  common  multiple  of  2  and  3.  At  this  point 
the  pupil  can  hardly  fail  to  see  that  the  second  common 
multiple  is  Q-\-(S^  the  third  is  6  +  6  +  6,  etc. ;  in  other 
words,  that  all  the  common  multiples  of  two  numbers 
are  formed  by  repeating  as  an  addend  the  least  common 
multiple.  He  can  then  be  led  to  see  the  reason  for  this, 
viz.,  that  (referring  to  the  foregoing  exam{)lc),  in  order 
to  get  a  common  multiple  of  2  and  3  larger  tlian  6,  it 
Avill  be  necessary  to  add  to  6  a  common  multiple  of  2 
and  3,  so  that  6  is  the  least  number  that  can  be  used. 

Numhers  Prime  to  Each  Other. — A  necessary  step 
preUminary  to  teaching  the  formal  process  is  the  bring- 
ing out  of  the  fact  that  the  least  common  multiple  of 
two  numbers  prime  to  each  other  is  their  product.  For 
example,  take  the  numbers  5  and  T :  a  common  multiple 
must  have  5  as  a  factor  and  7  as  a  factor ;  it  is,  there- 
fon^  5  niulti})lied  by  another  factor.  But  since  the  mul- 
tiple contains  7,  and  since  7  is  prime  to  5,  the  other  fac- 
tor of  the  multiple  must  contain  7.  Hence,  since  tlie 
smallest  multiple  of  7  is  7,  the  least  common  multiple 
of  5  and  7  is  5  X  7.  Similarly,  a  common  multiple  of 
4  and  9  is  a  multiple  of  4,  and  is  therefore  4  multiplied 


f)>fM 


'^'W 


b 


<tl 


mi' 


■  ■!■■  i 

lw{ 

$!■ 

rf 

ifl 

'.r'r. 

^w 

.'^ 

236 


THE  PSYCHOLOGY  OF  NUMBER. 


by  another  factor ;  but  the  common  multiple  is  a  mul- 
tiple of  9,  and  therefore,  since  4  contains  no  factor  of 
9,  the  other  factor  must  contain  9  ;  thus,  since  9  id  the 
least  number  that  will  contain  9,  the  least  common  mul- 
tiple of  4  and  9  is  4  X  9.  This  fact  has  generally  been 
taken  as  self-evident.  It  seems  best,  however,  to  help 
the  pupils  to  get  full  possession  of  it,  and  the  somewhat 
abstract  discussion  should  be  illustrated  by  a  series  of 
multiples  of  the  numbers 

5,  10,  15,  20,  25,  30,  35, 
Y,  14,  21,  28,  35,  42,  49. 

Here  30,  for  example,  can  not  be  a  multiple  of  7,  being 
made  up  of  the  factors  5,  2,  and  3,  all  of  which  are  prime 
to  7. 

The  next  step  will  be  to  find  the  least  common  mul- 
tiple of  two  numbers  not  prime  to  each  other,  say  36 
and  48.     Kesolving  each  of  these  into  simple  factors, 

we  see  that 

36  =  2X2X3X3 

48  =  2X2X2X2X3 

Thus  a  common  multiple  must  have  2  as  a  factor  four 
times,  and  3  as  a  factor  two  times,  so  that  the  least  com- 
mon multiple  is 

2X2X2X2X3X3,  or  144 

It  will  now  be  easy  to  bring  out  the  relation  of  this  to 
the  following  process : 

2)36,  48 

2)18,  24 

3)9,  12 

3,  4 


' 


MEASURES  AND  MULTIPLES. 


237 


The  first  division  shows  that  36  =  2  X  18,  and  48  = 
2  X  24.  Thus  a  common  mnkiple  must  have  2  as 
a  factor,  and  its  remaining  factors  must  make  up  a 
number  which  will  contain  exactly  the  numbers  18 
and  24.  Similarly,  a  common  multiple  of  18  and  24 
must  have  the  factor  2,  and  its  remaining  factors  must 
make  up  a  number  which  will  contain  exactly  tlie  num- 
bers 9  and  12,  etc.  Clearly,  then,  a  conimon  multiple 
must  have  tLj  factors  2X2X3,  and  its  remaining 
factors  must  constitute  a  number  that  will  contain  ex- 
actly 3  and  4,  two  numbers  prime  to  each  other,  and 
having  therefore  4x3  for  least  common  multiple. 
Hence  the  least  common  multiple  of  the  given  num- 
bers is 

2  X  2  X  3  X  3  X  4,  or  144 

It  will  now  be  easily  seen  that  it  would  have  been 
allowable  to  divide  at  once  by  12,  the  greatest  number 
that  will  divide  each  of  36  and  48,  and  the  pupil  can 
deduce  a  definite  method  of  finding  the  least  common 
multiple  of  any  two  given  numbers,  whether  he  can 
discover  factors  at  once,  or  is  compelled  to  resort  to 
the  process  of  finding  the  greatest  common  measure  of 
the  two  numbers. 

The  process  may  now  be  extended  to  the  case  of 
three  or  more  numbers  : 

2)12,  15,  18 

3)6,  15,     9 

2,     5,     3 

The  least  common  m.ultiple  must  have  the  factor  2,  and 
must  further  provide  for  the  numbers  6,  15,  9 ;  to  do 
this,  an  additional  factor,  3,  must  be  introduced,  and  it 


v. 


.'•(,-■' 


... .  i 


m : 


1 


(! 

1,1;! :  !     : 

li' 

-'I 
■.' 


f 


It 


^! 

i 
i 

I' 

1,   ; 

238 


THE  PSYCHOLOGY  OF  NUMBER, 


remains  yet  to  provide  for  the  numbers  2,  5,  3.  These 
latter  are  prime  to  one  another,  so  that  the  least  number 
that  will  contain  them  is  2  X  5  X  3 ;  and  hence  the  least 
common  multiple  of  12,  15,  18  is 

2  X  3  X  2  X  5  X  3,  or  180 

The  preceding  example  might  have  been  treated  as 

follows : 

6)12,  15,  18 

3)2,  15,     3 

2,     5,     1 

and  least  common  multiple  =  6X3x2X5  =  1 80. 
Here  the  lirs^  division  was  by  6,  which  is  seen  to  divide 
12  and  18.     But  the  division  by  a  composite  number 
may  lead  to  a  multiple  which  is  not  the  least. 

Example :  3)6,  15.  42 

2)2,     5,  14 


5, 


.  • .  L.  C.  M.  =  3  X  2  X  5  X  7  =  210 

If  6  had  been  taken  as  the  first  divisor,  the  w^ork  would 
have  stood  thus : 

6)6,  15.  42 
1,  15,     7 

and  the  result  w^ould  have  been 

6X15X7  =  3X2X3X5X7  =  630 

The  reason  for  the  difference  in  results  is  apparent. 
In  the  latter  process,  after  the  factor  6  of  the  least 
common  multiple  was  found,  it  was  supposed  that  there 
was  still  need  to  provide  for  all  the  factors  of  15 — i.  e., 
for  3  and  5 ;  whereas  the  factor  6,  already  taken,  con- 


MEASURES  AND  MULTIPLES. 


231) 


'iii 


tains  the  factor  3.  It  will  also  be  seen  wliv,  in  the 
case  of  jinding  the  least  common  nuiltiple  of  12,  15, 
and  18,  a  similar  way  of  proceeding  gave  the  coi'rect 
result.  The  pnpil  is  now  led  to  see  that  in  finding 
the  least  common  multiple  of  several  numbers  it  is 
frecpiently  necessary,  and  therefore  (practically)  always 
advisable,  to  divide  out  by  factors  that  are  prime 
numbers. 

Larger  Numbers. — The  thought  which  from  the  first 
has  directed  this  presentation,  and  which  is  now  clearly 
understood  by  the  pupils  is,  the  least  common  multiple 
of  several  numbers  must  contain  (1)  all  the  different 
factors  in  the  numbers,  (2)  each  in  the  highest  ])ower 
it  has  in  any  of  the  numbers.  The  pupils  will  now 
easily  deduce  the  rule  for  applying  the  greatest  com- 
mon measure  to  find  the  least  common  multiple  of 
numbers  that  can  not  be  resolved  by  inspection.  If 
the  least  common  multiple  of  851  and  1073  is  re<piired, 
they  will  find,  as  before,  851  =  23  X  37,  1073  ==  20  X 
37,  and  will  see  at  once  that  therefore  the  least  conunon 
multiple  is  23  X  29  X  37.  On  comparing  this  with 
the  two  numbers  and  their  greatest  conmion  measure 
they  will  readily  see  that  23  =  -\^—  =  ^i^\  and  that 

21)  —  -j'f^  =-§7^'  ^^^^  ^^^^^  therefore  the  least  com- 
mon multiple  of  two  numbers  may  be  found  by  divid- 
ing either  of  them  by  their  greatest  common  measure 
and  multiplying  the  quotient  by  the  other.  This  may, 
of  course,  be  readily  applied  to  all  cases;  for  example, 
find  the  least  common  multiple  of  12,  15,  18.  The 
greatest  common  measure  of  12X15  is  4;  .'.  their 
least  common  multiple  is  4  X  IT)  =  60  ;  the  greatest 
common  measure  of  00  and  18  is  0  ;  .  • .  the  least  coni- 
17 


h 


f 


UTTrW 


M 


240 


THE   PSYCHOLOGY  OF   NUMBER. 


111011   multiple  is  -^  X  18  =  180.     Similarly,  find  the 
least  common  multiple  of  12,  20,  36,  60,  54. 

The  least  common  multiple  of  12  and  20  is    60. 

''  "         60  and  36  is  180. 

"  "  "       180  and  54  is  540. 

Speaking  from  actual  experience,  we  have  not  a  doubt 
that  pupils  can  be  led  to  the  statement  of  the  principles 
and  rules  in  general  forms  :  If  a  —  7/ic,  and  h  =  no, 
when  c  is  the  greatest  common  measure  of  two  quan- 
tities a  and  i,  then  the  least  common  multiple  of  a  and 

h  is  innc,  which  is  ^  X  nc.  or  ^  X  'i^c.  etc. 

In  closing  this  chapter,  it  is  scarcely  necessary  to 
emphasize  the  importance  of  adding  an  interest  to  this 
part  of  the  work  in  arithmetic  by  drawing  upon  the 
great  variety  of  practical  problems  as  illustrating  meas- 
ures and  multiples ;  or  to  speak  of  the  need  of  having 
the  pupils  give  the  reasons  which  led  them  to  conclude 
that  the  problem  was  concerned  with  the  finding  of  a 
measure  or  a  multiple,  as  the  case  may  be. 


■s:i' 


1 , 

1 

1  ' , 

i 

■ 

li    A    :-& 

nd  the 


60. 
80. 
iO. 

I  doubt 
nciples 

>  quan- 
P  a  and 

sary  to 
to  this 
lon  the 
y  meas- 
having 
mclude 
\g  of  a 


CIIAPTEE  XIII. 


FRACTIONS. 


After  the  previous  discussion  on  the  nature  of  frac- 
tions (see  especially  Chapter  VII)  and  their  psychological 
relation  with  the  fundamental  operations,  a  brief  ref- 
erence to  some  of  the  points  brought  out  is  all  that 
is  needed  as  an  introduction  to  the  formal  teaching  of 
fractions. 

dumber  depends  upon  measurement  of  quantity. 
This  measurement  bemns  with  the  use  of  inexact  units 
— the  counting  of  like  things — and  gives  rise  to  addition 
and  subtraction.  From  this  first  crude  measurement  is 
evolved  the  higher  stage  in  wdiich  exactly  defined  nnits 
of  measure  are  used,  and  in  which  multiplication,  di- 
vision, and  fractions  arise.  Multiplication  and  division 
brins:  out  more  clearly  the  idea  of  number  as  measure- 
ment  of  quantity — as  denoting,  that  is,  (?')  a  unit  of 
measure  and  (/«)  times  of  its  repetition.  The  fraction 
carries  the  development  of  the  measuring  idea  a  step 
further.  As  a  mental  process  it  constitutes  a  more 
definite  measurement  by  consciously  using  a  defined 
unit  of  measure ;  and  as  a  flotation,  it  gives  complete  ex- 
pression to  this  more  definite  mental  process.  Frac- 
tions therefore  employ  more  explicitly  both  the  con- 
ceptions involved  in  multiplication  and  division — name- 

241 


n 


im^ 


II     4    V  •' 


). 


I 


li;:  ^ 

115   %. 


■'I 


I. 

u 


-'! 

1    i 

1 

.1 

■1 

liki 

tJi 

2i2 


THE   PSYCHOLOGY   OF  NUMBER. 


ly,  analysis  of  a  whole  into  exact  units,  and  synthesis  of 
these  into  a  defined  whole.  The  idea  of  fractions  is 
present  from  the  first,  because  division  and  multiplica- 
tion are  implied  from  the  first.  There  is  no  number 
without  measurement,  nor  measurement  without  frac- 
tions. Even  in  whole  numbers,  as  has  been  pointed  out, 
both  "  terms  "  of  a  fraction  are  implied  in  the  accurate 
interpretation  of  the  measured  quantities. 

Since  there  is  nothing  new  in  the  process  of  frac- 
tions, so  in  the  teaching  of  fractions  there  is  nothing 
essentially  different  from  the  familiar  operations  with 
whole  numbers.  If  the  idea  of  number  as  measurement 
has  been  made  the  basis  of  method  in  primary  work 
and  in  the  fundamental  operations,  the  fraction  idea 
must  have  been  constantly  used,  and  there  is  absolutely 
no  break  when  the  pupil  comes  to  the  formal  study  of 
fractions.  There  is  only  before  him  the  easy  task  of 
examining  somewhat  more  attentively  the  nature  of  the 
processes  he  has  long  been  using.  The  suggestions 
made  in  reference  to  primary  teaching  and  formal  in- 
struction in  the  fundamental  operations  apply  with 
equal  force  to  the  teaching  of  fractions.  The  meas- 
uring idea  is  to  be  kept  prominent :  avoidance  of  the 
fixed  unit  fallacy  and  its  logical  outgrowth,  the  use 
of  the  undefined  qualitative  unit — the  pie  and  apple 
method — as  the  basis  for  developing  "  fractional "  units 
of  measure,  and  the  "  properties  of  fractions  "  ;  the  es- 
sential property  of  the  imit  in  measurement — the  7neas- 
ured  part  of  a  measured  whole  ;  the  logical  and  psycho- 
logical relation  between  the  niimher  that  defines  the 
measuring  unit  and  the  number  that  defines  the  meas- 
ured  quantity  j    or  as  it  is  sometimes  expressed,  the 


FRACTIONS. 


213 


Lidy  of 


"  relation  between  the  size  of  tlie  parts  "  (the  measur- 
ing^ units)  and  the  numher  of  tlie  parts  composing  or 
e(|ualling  the  measured  (juantity ;  these  and  all  kindred 
})uints  that  have  been  brought  out  in  discussing  number 
as  measurement,  and  numerical  operations  as  simply 
})hases  in  the  development  of  the  measuring  idea,  can 
not  be  ignored  in  the  teaching  of  fractions,  because  they 
can  not  be  ignored  in  the  teaching  of  whole  numbers. 
Exact  number  demands  definition  of  the  unit  of  meas- 
ure ;  the  fraction  completely  satisfies  this  demand  by 
stating  or  defining  expressly  the  unit  of  measure.  In 
all  number  as  representing  measured  quantity  the  ques- 
tions are  :  What  is  the  unit  of  measure,  and  how  is  it 
defined  or  measured  ?  How  many  units  equal  or  con- 
stitute the  qaantity  %  These  questions  only  number  in 
its  fractional  form  completely  answers.  It  is  the  com- 
pletion of  the  josychical  process  of  number  as  measure- 
ment of  quantity  ;  the  idea  of  the  quantity  is  made 
definite,  and  it  is  definitely  expressed. 

While  the  following  treatment  of  fractions  is  in 
strict  line  with  the  principles  of  number  set  forth  in 
these  pages,  and  has  stood  the  test  of  actual  experience, 
it  is  given  only  by  way  of  suggestion.  The  principles 
are  universal  and  necessary  ;  devices  for  their  effective 
application  are  within  certain  limits  individual  and  con- 
tingent. Principles  are  determined  by  philosophy,  de- 
vices by  rational  experience.  The  teacher  must  be  loyal 
to  princi})les,  but  the  slave  of  no  man's  devices. 

1.  The  Function  of  the  Fraction. 

1.  In  its  primary  conception  a  fraction  may  be  con- 
sidered as  a  number  in  which  the  unit  of  measure  is 


-i 


I 

i 

1 1 
r 

II  I 


'  } 


'<}'  '• 


Ih 


i.! 


iiU, 


1: 


244 


THE  PSYCHOLOGY  OF  NUMBER. 


expressly  defined.  In  the  quantities  4  dimes,  5  inches, 
9  ounces,  the  units  of  measure  are  not  explicitly  defined  ; 
their  value  is,  however,  implied,  or  else  there  is  not  a 
definite  conception  of  the  quantity.  In  $-3^,  -f-^  foot, 
-j^  pound,  the  units  of  measure  are  explicitly  defined  ; 
and  each  of  these  expressions  denotes  four  things  :  1.  The 
unity  (or  standard)  of  reference  from  which  the  actual 
unit  of  measure  is  derived.  2.  IIow  this  unit  is  de- 
rived from  the  unity  of  reference.      3.   The  absolute 

■y 

number  of  these  derived  units  in  the  quantity.  4.  This 
number  is  the  ratio  of  the  given  quantity  to  the  unity 
of  reference. 

For  example,  in  -^-q  pound,  the  unity  of  reference  is 
one  pound  ;  it  is  divided  into  sixteen  equal  parts,  to  give 
the  direct  measuring  unit ;  the  number  of  these  units 
in  the  given  quantity  is  nine;  the  ratio  of  the  given 
quantity  to  the  unity  of  reference  is  nine. 

2.  If  properly  taught,  the  pu])il  knows — if  not,  he 
must  be  made  to  know — that  any  quantity  can  be 
divided  into  2,  3,  4,  5  .  .  .  n  ecpial  parts,  and  can  be 
expressed  in  the  forms  f,  f,  f,  |-  •  •  •  ^-  Familiar 
with  the  ideas  of  division  and  nnilti plication  which  be- 
come explicit  in  fractions,  he  learns  in  a  few  niinutes 
(has  already  learned,  if  he  has  been  rationally  instructed) 
that  any  quantity  may  be  measured  by  2  halves,  or  3 
thirds,  or  4  fourths  ...  or  ?2  7iths ;  that  to  take  a  half, 
a  third,  a  fourth  ...  an  7?^th  of  any  quantity,  it  is  only 
necessary  to  divide  by  2,  or  3,  or  4  .  .  .  or  ?i ;  that  if, 
for  example,  16  cents,  or  16  feet,  or  16  pounds  has  been 
divided  into  four  parts,  the  counts  of  the  units  in  each 
case  are  one,  two,  three,  four,  or  one  fourth,  two  fourths, 
three  fourths,  four  fourths  ;  that  each  of  these  units — 


L 

<M_, 

FRACTIONS. 


245 


not  a 


unity 


or  ?> 


fourths — is  measured  by  otlier  units,  and  can  be  ex- 
pressed as  integers,  namely,  4  cents,  4  feet,  4  pounds, 
and  so  on,  with  kindred  ideas  and  operations. 

3.  The  Primary  Practical  Prhiciple  in  Fractions. 
— It  is  clear  that  this  complete  exj^ression  for  the  num- 
ber process  is  the  fundamental  principle  employed  in  the 
treatment  of  fractions :  if  both  terms  of  a  fraction  be 
multiplied  or  divided  by  the  same  nund)er,  the  numer- 
ical value  of  the  fraction  will  not  be  changed.  This 
principle  is  usually  "  demonstrated " ;  it  is,  however, 
involved  in  the  very  conception  of  number,  and  seems 
as  difficult  to  demonstrate  as  the  delinition  of  a  triangle  ; 
but  intuitions  and  illustrations  to  any  extent  may  be 
given.  Any  12-unit  quantity,  for  example,  is  measured 
by  f ,  f ,  -}-|,  or  by  f ,  f ,  \\ ;  the  identity  of  the  quan- 
tity remains  unchanged  in  the  changing  measurements. 
Moreover,  if  half  the  quantity  be  measured,  the  identity 
of  J,  f,  f,  3^  is  seen  at  once.  The  principle  is,  of 
course,  that  in  a  given  measured  quantity  the  "  size " 
of  the  units  varies  inversely  witli  their  number.  This 
principle  is  said  to  be  beyond  the  comprebension  of  the 
pupil.  On  the  contrary,  if  constructive  exercises,  such 
as  have  been  described,  have  been  practised,  there  comes 
in  good  time  a  complete  recognition  of  the  principle. 
When,  for  instance,  the  child  measures  off  any  24-unit 
quantity  by  twos,  threes,  fours,  sixes,  eights,  lie  can  not 
help  feeling  the  relation  between  the  magnitude  and 
the  number  of  the  measuring  parts.  This  is,  in  fact, 
the  process  of  number. 

Proof  of  the  Principle. — If  the  first  vague  aware- 
ness of  the  relation  does  not  grow  into  a  clear  compre- 
hension of  it,  clearly  the  method  is  at  fault.     In  any 


■ii 


,''f,  -^ 


Hi 


!!■ 


jj^ » 

ir^«" ' 

'ii'  ' 

i; 

*:  ■     • 
1  ( 

( 
I 

•1 

.'    i 


A 


240 


THE  PSYCHOLOGY  OF   NUMBER. 


ease,  if  the  pupil  does  not  understand  the  principle 
after  rationally  using  it,  any  formal  "  demonstration  '* 
is  a  mere  delusion  ;  for  anv  so-called  demonstration  is 
grounded  on  the  principle — in  general  is  the  principle 
— merely  illustrated  or  used  in  a  disguised  form.  For 
example  :  Prove  l^^f  =  ^l-J-.  Since  $|-  =  ,^ J  x  3  :  mul- 
tii)ly  by  4,  and  we  have  8f  X  4  =  s^  X  4  X  3  =  ^3 ; 
multiply  these  equals  by  5  : 

.  • .   si^f  X  4  X  5  =  §3  X  5  =  $15  ;   but   also  $fg-  X 
20  =  $15; 

.  • .  $f  X  20  =  $^1^  X  20;  dividing  equals  by  equals; 

.  • .    $1  =  $JJ^     Or,  generally  :   ^  X  h  =  a',   multi- 
ply both  sides  by  n. 

.  • .  ^  X  nh  —  na ;  but  ^X  nh  =  na ; 

.  • .  |-  X  nh  =  '^X  nh;  dividing  both  sides  by  nh ; 


n 
T 


nn 
nb' 


These  and  similar  proofs  are  in  essence  the  idea  already 
considered  :  that  if  a  quantity  is  divided  into  a  certain 
number  of  equal  parts,  each  part  has  a  certain  value ; 
if  into  twice  the  number  of  parts,  each  part  has  half 
the  value  of  the  former  part ;  if  into  three  times  as 
many  parts,  each  has  a  third  of  the  value;  if  into  n 
times  as  many  parts,  each  has  1  nth  of  the  value. 

If  formal  proof  is  wanted  of  this  important  princi- 
ple (which  is,  onco  more,  the  principle  of  number),  the 
following  is  perhaps  as  intelligible  as  any  other.  To 
prove,  for  example,  that  $|  =  $|^-  =  $jV^  —^  etc.,  we 

tCl  —  09     —  __2  5_  —    pf  p 
.       C>H  —  C:3  7   —  C<  7  5,  —    pfp 


<1\    I 

^1  I 


FRACTIONS. 


24: 


ix 


4.  The  Fraction  as  Division. — AVliile  in  its  primai-v 
concoption  tlie  fraction  if  not  simj^ly  a  formal  division, 
it  nevertlieless  involves  the  idea  of  division,  and  can  not 
be  fully  treated  without  identifvin<]^  it  with  the  formal 
process.  The  quantity  -j^^  foot,  first  re<rarded  as  -j^ 
foot  X  7,  must  be  recognized  in  its  psycliological  cor- 
relate, 7  feet  X  iV—i- ^^v  7  feet -^  12.  As  has  been 
shown  more  than  once,  these  measurements  can  not 
but  be  recognised  as  two  phases  of  the  same  measure- 
ment, whenever  the  process  becomes  the  object  of  con- 
scious attention.  It  is  the  law  of  commutation,  the  con- 
nection between  the  number  and  the  magnitude  of  the 
units  in  a  measured  quantity.  If  we  do  not  know  that 
J-  of  3  times  a  quantity  is  3  times  ^  of  the  quantity — 
or,  generally,  that  ^  of  n  times  q^n  times  ^  of  »/— we 
have  no  clear  conception  of  number.  If  a  quantity  is 
measured  by  -^  of  a  certain  unity  of  reference  taken  7 
times,  this  is  seen  to  be  identical  with  -^^  of  one  of 
these  unities  -J-  ^V  of  a  second  +  tV  ^^  ^  i\iiv^  •  •  • ;  that 
is,  in  all,  J^  of  seven  of  them. 

Numerous  illustrations  and  so-called  proofs  may  be 
given.     Examples : 

(1.)  Show  that  I  of  any  quantity  is  equal  to  J  of  3 

I  times  the  quantity.    Let 

A ^-^ ' Li? j3    ^'i]    ])e    any   quantity 


c- 

E  — 


H 


T. 


D    whatever,  measured   in 
P    fourths    and    expressed 
as  1^  ;    and   C  D,  E  F 


each  measured  quantities  equal  to  A  B.  It  is  obvious 
that  A  K,  which  is  J  of  A  B,  is  equal  to  A  G  +  C  II  + 
E  L  ;  that  is,  ecpial  to  J-  of  3  times  A  B. 

(2)  To  show  that  yV  foot  X  7  =  7  feet  X  tV— i-  e., 


t   U        .1 


1 1  :  J    i'  >  ■     t 


m  > 


Ml'  » 


1,1 


:  '^  ? 


!•■ 


-  !i  ' 


m 


ill 


I'Hi 


1: 


248 


THE   PSYCHOLOGY   OF   NUMBER. 


=  7  feet  -T-  12.  The  unit  of  reference,  1  foot,  may  bo 
thought  of  and  expressed  as  12  twelfths  foot : 

.  • .  7  feet  =  84  twelfths  foot ; 

.  • .  7  feet  X  iV  =    "^  twelfths  foot  =  J^-  foot  X  7. 

(3)  To  prove  that  SJ  X  3  is  equal  to  $3  X  i : 

4  times  $J  ==  ^1  ;  multiply  these  equals  by  3 ; 
.  • .  4  times  ^J  X  3  =  S3  ; 

. ' .  8i  X  3  =  $3  -r-  4.  Or,  using  q  for  any  quantity, 
4  times  ^q  =  q\ 

.  • .  4  times  ^^^  X  3  =  3^' ;  hence,  J*/  X  3  =  3^/  -^  4. 

(4)  Or,  generally,  ~qXrri  =  iiiq  -r-  n.     For 
n  times  ^  ^  z=  (^  ; 

,' .  n  times  ~q  X  fn  =  mq ; 

j^q  X  m  =  mq  -i-  n. 

Such  formal  proofs  are  useful  and  even  necessary,  but 
are  likely  to  be  misleading  unless  the  pupil  has  evolved, 
from  rational  use  of  the  principle,  a  clear  idea  of  the 
relation  between  times  and  parts,  the  importance  of 
which  has  been  emphasized  in  this  book ;  he  is  apt  to 
become  a  mere  spectator  in  the  manipulation  of  sym- 
bols, rather  than  a  conscious  actor  in  the  mental  move- 
ment which  leads  to  complete  possession  of  the  thought. 

II.  Change  of  Form  in  Fractions. 

1.  From  what  has  been  already  said,  it  appears  that 
any  quantity  may  be  expressed  in  the  form  of  a  frac- 
tion having  any  required  denominator.  Express  9  yards 
in  eighths  of  a  yard.     Since  the  unit  of  measure  is  f ,  9 


FRACTIONS. 


2-19 


Piich  nnits  is  ■^,^^-.  Similarly,  >«^7  expressed  «as  liundredtlis 
is  -J^^,  etc.  In  gciieml,  any  qiuintity  of  q  units  of  meas- 
ure exiiressed  as  ?/tlis  is  -"-. 

2.  In  the  same  way,  any  quantity  expressed  in  frac- 
tional form  may  be  changed  to  an  e([uivalent  fraction 
having  any  denominator.  Transform  Jr'|  into  an  e([niva- 
lent  fraction  having  denominator  2(J.  We  can  follow 
either  of  two  plans  : 

(1)  20  is  a  multij)le  of  5  by  4  ;  we  therefore  mulri- 
])ly  both  terms  of  the  given  fraction  by  4,  getting  s|5-. 
This  is  best  in  prac^tice. 

(2)  Since  the  new  denominator  is  to  be  20,  m'C 
regard  the  unit  of  measure  as  'tlJ-;  ^|  is  ^i  X -i ; 
but  ^  =  one  fifth  of  Sf  J-  =  8/o  ; 


S4 

"0 


—   K  1  (5 


It  may  be  remarked  that  in  such  transformations  the 
new  denominator  is  generally  a  multi])le  of  the  original 
denominator.  If  it  is  not,  the  new  equivalent  fraction 
will  be  complex,  it  will  have  a  fractional  numerator. 
Thus,  if  it  is  re([uired  to  transform  ^  yard  to  an  equiva- 
lent fraction  with  denominator  12,  we  niultiply  both 


•  '  a 


terms  by  1^,  with  the  result  .^*^. 

3.  It  is  often  necessary  or  convenient  to  reduce  a 
fraction  to  its  lowest  terms — that  is,  to  express  it  in  terms 
of  the  largest  unit  of  measure  as  defined  by  the  unity 
of  reference.  This  is  done  by  dividing  both  terms  of 
the  fraction  by  their  greatest  common  measure ;  thus, 
^tV  ^^'^  e(|uivalent  to  %^,  in  which  the  quantity  is  ex- 
pressed in  the  largest  unit  of  measure,  as  delined  by 
the  unity  of  reference,  the  dollar.     The  principle  in- 


m 


.1  i 


^*"^^ 


'If  i 


250 


THE  PSYCHOLOGY  OF   NUMBER. 


volved  is  that  stated  in  I,  3 — viz.,  the  numerical  value 
of  the  units  is  increased  a  number  of  times,  tlie  number 
of  them  is  diminished  the  same  number  of  times. 

In  practice,  the  greatest  common  measure  can  gener- 
ally be  foiMid  by  inspection,  as  described  in  Chapter  XII. 
612  ^  2  X  2  X  3  X  >3  X  17  _  17 
~  19* 


Thus, 


In  some  cases 


'  ()84      2  X  2  X  3  X  3  X  ID 

the  greatest  common  measure  must  be  found  by  the 

general  method  described  in  the  same  chapter.     Thus, 

79409 
if  the  proposed  fraction  is  ^rrr^r—-, ,  we  should  discover 
^  z()obOo 

the  greatesi:  common  divisor  to  be  18 ;  en  dividing  both 

P1 1  ^ 
terms  of  the  given  fraction  there  results  ^.-7777  ,  which 

is  the  simplest  of  all  equivalent  fractions. 

4.  In  changing  a  mixed  number  to  an  improper  frac- 
tion, and  vice  versa,  the  primary  principle  of  fractions 
applies  at  once : 

(1)  Reduce  75f  yards  to  an  improper  fraction.  The 
expression  =  75  yards  -|-  f  yard  ;  express  75  yards  in 
form  of  a  fraction  with  denominator  3  : 

1  yard  is  J  yard  ; 
75  yards  is  |-  yard  X  75  =  ^^  yards ; 
.  • .  75  yards  +  f  yard  =  ^l^^-  yards  =  ^^  yards. 

(2)  In  the  converse  operation  either  consider  the 
problem  as  a  case  of  formal  division  giving  75f ,  or  con- 
sider the  expression  as  denoting  so  many  thirds  of  a. 
yard  ;  then  3-thirds  —  one  yard  ;  how  many  3-thirds  in 
227  thirds?  Evidently,  as  before,  a  case  of  division, 
pving  75  ones  and  two-thirds  remainder — that  is,  75J 
yards.  It  may  be  observed  that  in  (1),  while  tht^  ])ri- 
mary  measurement  of  the  qnantity  is  3  units  X  75  -|-  2 


FRACTIONS. 


251 


ds. 
the 
con- 
of  a 
Is  in 
sioii, 

pri- 

+  2 


I 


units,  and  we  multiply  3  by  75,  it  is  equally  lop;ic'al  to 
use  the  correlate  75  units  X  3.     (See  pa<ije  77.) 

5.  Comparison  of  Fractions.  Common  Denomina- 
tor.— It  is  often  ^lecessary  to  transform  fractions  liaving 
different  denominators  to  equivalent  fractions  with  the 
same  denominator.  Quantities  can  not  be  deiinitely  com- 
pared, we  have  seen,  unless  they  have  the  sam'^  unit  of 
measure.  We  can  not  compare  directly  5  feet  and  5 
rods ;  they  must  both  be  expressed  in  terms  of  the  same 
measuring  unit.  So  with  4  dollars  and  7  uimes.  Tn  like 
manner,  J^f  and  S^|-  can  not  be  definitely  compared  ;  they 
have  a  common  (primary)  unit  of  reference,  but  not  a 
common  (actual)  unit  of  measure  ;  but  they  can  both 
be  expressed  in  terms  of  a  common  unit  of  measure. 
They  can  be  expressed  as  j^-|-|-,  ^|-|,  or  ^ffj,  ^f o^^  ^^^• 
For  comparison,  it  is  generally  most  convenient  to  ex- 
press such  quantities  in  terms  of  the  greatest  common 
unit  of  measure,  and  this  is  determined  by  the  least 
conimon  divisor  or  denominator. 

(1)  (/ompare  the  quantities  f  foot  and  -S-  foot.  Here 
the  least  common  multiple  of  the  denominators  is  42, 
and  tl\e  fractions  become  f f  foot  and  fj  foot ;  the  lat- 
ter fraction  is  therefore  the  greater. 

(2)  Which  is  the  greater.  J^Jf  or  $y%  ?  The  least 
coimnon  denominator  is  128,  and  tiie  128th  of  a  dollar 
is  then  the  common  unit  of  measure ;  the  fractions  are 

therefore  'j^  y/g-,  '^iVs  ,  ^^"^^  hence  the  former  is  the 
greater. 

It  is  \^  bo  observed  that  if  the  (piestion  is  simply, 
Which  is  the  greatest  of  a  number  of  fractions'^  it  can 
be  answered  l)y  reducing  them,  by  a  similar  process,  to 
a  common  numerator. 


,Ji,  '.'iHlliI    , 


l!''' 


III*' 


H    1' 


I 


Nil  ft  i 


( , 


252 


THE  PSYCHOLOGY  OP  NUMBER. 


(1)  AVIiicli  is  the  greater,  $f  or  $f  ?  Tlie  least  oom- 
iiion  multiple  of  the  numerators  is  30  ;  multiplying  both 
terms  of  the  first  fraction  by  6,  and  of  the  second  by  5, 
we  have  $j^,  '^|-§-;  and  since  the  latter  has  the  smaller 
denominator  (therefore  greater  unit  of  measure),  it  is  the 
larger  fraction. 

(2)  Compare  the  quantities  $f ,  Sy^^-,  $-|,  by  reduc- 
ing them  to  a  connnon  numerator.  The  least  common 
multiple  of  3,  7,  5  is  105  ;  the  multi})liers  for  the  terms 
of  the  respective  fractions  are,  therefore,  35,  15,  21, 
giving  ^§-,  iff,  i-|-| ;  hence,  the  first  fraction  is  the 
greatest,  and  the  second  one  the  least.  In  this  case  the 
comparison  would  be  easier  by  reducing  to  a  common 
denominator. 

The  Greatest  Commoii  Pleasure  and  Least  Common 
Multiple  of  Fractions. —  Here,  also,  there  is  notliing 
essentially  different  from  the  corresponding  operations 
in  whole  numbers.  As  has  been  said,  quantities,  in 
order  to  be  compared,  must  be  expressed  in  terms  of 
the  same  unit  of  measure.  If  fractions  have  a  conunon 
denominator — represent,  that  is,  (piantities  defined  by 
the  same  unit  of  measure — the  ordinary  rules  for  meas- 
ures and  multiples  at  once  apply.     Examples  : 

(1)  Find  the  greptest"  common  measure  of  \  yard 
and  "I  yard.  Expressed  with  the  same  denominator 
these  become  fj  yard,  W  yard.  The  greatest  conunon 
measure  of  30  and  16  is  2 ;  therefore  -^^  is  the  greatest 
common  measure  required. 

(2)  What  is  the  greatest  length  that  is  contained  an 
integral  number  of  times  in  1S|  feet  and  57|^  feet? 
Change  to  inqiroper  fractions  with  least  common  de- 
nominator :  -ij^-,  ^-^^.     The  greatest  common  measure 


FRACTIONS. 


253 


meai 


c 


of  these  mimerators  is  23;  therefore  -f|,  or  2^^^,  is  the 
required  nimiber. 

(3)  Four  bells  begin  tolling  together ;  they  toll  at 
intervals  of  1,  1|-,  l^^^,  1  j^^^  seconds  respectively ;  after 
what  interval  will  they  toll  together  again  ? 

Here  the  least  common  multiple  of  the  numbers  is 
required.  Change  to  improper  fractions  with  least 
common  denominator  :  i||,  iff,  -ff^,  i|f.  The  least 
ommon  multiple  of  the  numbers  is  14,040,  and  the  least 
common  multiple  required  is  therefore  -L-}f§^  =  ir7; 
therefore  the  required  interval  is  117  seconds. 

III.  The  Fundamental  Operations. 

1.  Addition  of  Fractions. — (1)  When  the  fractions 
have  a  common  denominator — that  is,  when  they  de- 
note the  same  unit  of  measure — the  process  is  the  same  as 
in  whole  numbers.  There  is  no  essential  difference  be- 
tween the  operations  3  dimes  +  4  dimes  and  '^-f-^  -\-  %y^. 
The  only  dilference  is  in  the  mode  of  eir^n'essing  the 
unit  of  measure — seven  dimes  in  the  cue  case,  seven 
tenths  of  a  dollar  in  the  other. 

(2)  AVhen  the  denominators  are  different  the  frac- 
tions must  be  reduced  to  equivalent  fractions  having  a 
conmion  denominator — that  is,  they  must  be  expressed 
in  terms  of  a  common  unit  of  measure  (see  II,  G).  We 
can  not  add  5^4  and  4  half-eagles,  nor  4  feet  and  4  yards, 
till  we  express  the  quantities  in  a  common  unit  of  meas- 
ure, the  first  two  in  the  form  $4  + $20,  and  the  second 
in  the  form  4  feet  ~|- 12  feet.  So  we  can  not  add  f 
yard  and  f  yard  without  first  expressing  them  in  terms 
of  a  connnon  measuring  unit.  For  convenience  we 
select,  as  before  said,  the  gradtst  common  unit  of  meas- 


f 

}" 

)  ■ 

1 

\ 

1 

w  - 

1  ' 

1 

( 
1 

.1 

I" 


tS 


% 


25i 


THE   PSYCHOLOGY  OF  NUMBER. 


lire  as  delined  in  relation  to  the  primary  nnit  of  refer- 
ence, the  yard.  Tliis  greatest  niiit  is  given  by  the  least 
common  multij^le  of  the  denominators  4  and  9,  which 
is  36.  The  qnantities  expressed  in  the  connnon  nnit  of 
measure  are  therefore  f  J  yard  and  W  yard,  and  their 
sum  is  (27  +  16  =  43)  3(;ths  of  a  yard,  or  -JJ  yard. 
The  operation  is  essentially  the  same  as  that  of  iinding 
the  sum  (27  +  16)  inches. 

(3)  In  addition  of  mixed  numbers  it  is  best  to  find 
the  sum  of  the  whole  numbers,  the  sum  of  the  fractions, 
and  then  the  sum  of  these  two  results.  Improper  frac- 
tions should,  in  general,  be  expressed  as  mixed  numbers. 
Find  the  sum  of  13^  yards,  17|  yards,  Jj^- yards.     The 

sum  is 

lS4  +  lT|  +  a|  =  i:5  +  17  +  3  +  A+5  +  3 

oo  _|     16  '  30  ^  27 

r=  33  +  23V  -  053V  yards. 

2.  Suhtradion  of  l^raethms. — Since  subtraction  is 
the  inverse  of  addition,  the  san)e  ])rinciples  and  methods 
apply  in  both.  In  subtraction  of  mixed  nund)ers  it  is 
generally  best  first  to  change  the  fractional  ])arts  to 
equivalent  fractions  w^ith  the  same  unit  of  measure, 
rud  tlien  j^jrform  the  subtraction.     Example: 

(1)  How  much  was  left  of  $10j^  after  a  payment 

Expressing  the  fractional  parts  with  common  de- 
nominator : 


J!>10-V 


(2)  How  much  was  left  of  16f  yards  of  cloth  after 
6|  yards  :<.M;re  cut  from  it? 

deducing  the  fractional  parts  to  common  denomi- 
nator : 


m 


FRACTIONS. 


after 


255 


16| 


10      ,2 


(\8  —  f^64 


15 

A6  4 


72  +  27 


72 


^tI  y^^ds  remainder. 


Here  the  fractional  part  of  the  subtrahend  is  greater 
than  the  fractional  part  of  the  minuend  ;  the  minuend  is 
therefore  changed  to  the  form  15+1  +  f|  =  i^r2  +  2j  _ 


72 


1  n99 
10  J  J 


In  actual  work  we  may  take  64  from  72,  the 

denominator  of  the  minuend  fraction,  and  add  to  the 

remainder   the   numerator  of   the    minuend   fraction. 

Thus,  we  can  not  subtract  04  from  27 ;  we  subtract  it 

from  72,  and  add  the  remainder,  8,  to  27,  getting  -fl-. 

This  is  equivalent  to  taking  1  (=  ^)  from  the  minuend, 

and  uniting  it  with  the  minuend  fraction,  as  has  been 

done  in  the  example. 

3.  Multiplicatioii  of  Fractions.  —  (1)   When   the 

multiplicand  is  a  fraction  and  the  multiplier  a  whole 

number,  the  operation  is  exactly  like  multiplication  of 

integers.     To  find  the  cost  of  12  yards  of  cloth  at  $J 

a  yard,  we  multiply  3  by  12  and  define  the  product  by 

the  proper  unit  of  measure.     In  finding  the  cost  of  12 

yards  at  $3  a  yard,  the  complete  ]irocess  is  $1  X  3  X  12  ; 

we  operate  Avith  the  pure  numbers  3  and  12,  getting 

36,  and  define  the  product  by  naming  the  proper  unit 

of  measure  one  dollar  y  the  cost  is  then  36  dollars.     In 

the  proposed  case  we  do  exactly  the  same  thing  :  $J  X 

3  X  12 — that  is,  36  times  the  p!'oper  unit  of  measure 

($J) — and  the  product  is  36  qxicrter  dollars.     Neither 

the  process  nor  the  product  changes,  because  the  nnit^ 

or  the  manner  of  writing  it,  happens  to  change. 

(2)  AVhen  the  multiplier  is  a  fraction,  exactly  the 
18 


IH 


I 


/i.^^'l 


M^K" 


256 


THE  PSYCHOLOGY  OF  NUMBER. 


II  I' 


\ 


n. 


same  principles  hold  ;  in  fact,  the  measured  qnantity, 
$i  X  12,  is  identical  with  $12  X  f .  In  this  conception 
of  quantity  (money  value)  we  have  nothing  to  do  with 
yards,  and  either  form  of  the  measurement  may  be 
taken.  In  fact,  $i  X  12  is  f  of  $1  +  |  of  $1  +  J  of 
$1,  and  so  on  12  times — that  is,  f  of  $12.  The  multi- 
plicand is  always  a  unit  of  measure ;  the  multiplier 
always  shows  how  this  unit  is  treated  to  make  up  the 
measured  whole.  It  is  purely  an  operation.  In  this 
example  the  denominator  shows  how  the  unit  $12  is  to 
be  dealt  with  in  order  to  yield  the  derived  unit  of  meas- 
ure :  it  is  to  be  divided  into  four  parts,  and  the  derived 
unit  thus  found  is  to  be  taken  three  times.  As  already 
shown,  from  the  nature  of  the  fraction  it  denotes  three 
times  one  fourth  of  the  multiplicand,  or  one  fourth  of 
three  times  the  multiplicand — that  is,  $J^X  3,  or  $  ^  4—  • 

(3)  The  explanation  usually  given  of  the  process  is 
in  harmony  with  this.  This  explanation  considers  the 
multipT  and  as  a  case  of  pure  division  ;  that  is,  J  is  one 
fourth  of  3,  and  to  multiply  a  (juantity  by  f  is  to  take 
one  fourtii  of  3  times  the  quantity.  In  fact,  in  all 
operations  with  fractions  the  idea  of  division,  as  well 
as  of  multiplication,  is  present;  a  factor  and  a  divisor 
are  always  elements  in  the  problem. 

(4)  The  method  to  be  followed  when  both  factors 
are  of  fractional  form  involves  nothing  different  from 
the  other  two  cases. 

Tlie  price  of  f  yard  of  cloth  at  $f  a  yard  is  to  be 
found.  The  result  is  indicated  by  $f  X  4 ;  that  is,  as 
before,  4  times  a  certain  quantity  io  to  be  divided  by 
9,  or  \  of  the  quantity  is  to  be  multiplied  by  4.  In 
the  first  case,  $|  X  4  is     fher  ^J-  X  3  X  4  =  $i  X  12  = 


Hi    1 '  i 


FRACTIONS. 


257 


to  be 


IS,  as 


o 


$3  ;  or  8i  X  4  X  3  =  $1  X  3  =  $3  ;  and  -J  of  this  is 
$f ,  or  ^. 

It  may  be  ob?ervcd  tliat  we  may  change  the  multi- 
pheand  into  an  equivalent  fraction  with  a  miit  of  meas- 
ure determined  by  a  multiple  of  the  denominators.  In 
$1  X  i,  for  example,  we  have  ^}  Xi  =  $^X4:  =  ^  = 
$-|-.  Tiie  complete  process  is  seen  to  be  J  X  i  =  Jf . 
But  since  numerators  are  always  factors  of  a  dividend, 
and  denominators  factors  of  a  divisor,  common  factors 
may  be  divided  out.  In  $g\  X  4,  for  instance,  the  value 
of  the  quantity  is  the  same  whether  we  take  4  times  the 
number  of  units  (—  $g-|-),  or  make  the  units  -I  times  as 
large  ($-|).  Tliis  is  nothing  but  the  application  of  fun- 
damental principles  (see  page  22G)  of  multiplication  and 
division.  If  we  have  to  divide  210  by  21,  we  may  pro- 
ceed thus  :  -^V-  -  ^"3 1  y-  =  I  X  -?-  X  5  =  5.  Again, 
32310  -r-  385  ==  ^J±l^m^^^  =  ,1  X  H  X  I  X  r  X  2 
X  6  =  1  X  84:  =:.  84. 

4.  Divimm  of  Fractions. — (1)  When  the  divisor  is 
an  integer  and  the  dividend  a  fraction. 

Paid  %-^  for  5  yards  of  calico,  what  was  the  price 
per  yard  ?  One  yard  will  cost  one  fifth  of  %-^,  or  $/^. 
At  $5  a  yard,  how  much  lace  can  be  bouglit  for  ^-^-^  ? 
The  answer  is  indicated  in  $^  ---  §5  ;  the  quantities 
must  have  the  same  unit  of  measure,  and  the  expression 
is  equivalent  to  $-i^ -^  8t¥  ~  "^ -^^*^  =  A  5  ^^ence,  -^ 
yard  of  lace  can  be  bought. 

(2)  When  the  divisor  is  a  fraction,  and  the  dividend 
an  intesjer. 

At  8f  a  yard,  how  many  yards  of  dress  goods  can 
It  for  $6  % 
number  of  \ards  is  iriven  in  ^6  -~  tl.  where. 


be  bou 


given 


•'4' 


i  ■'■ 


mmf  ii     <  !■ 


''i'*> 

« 

t 

r  , 

'. 

;  ■^■'^'' 

'« 

f 

i: 

1 1)  ^ 

:'!; 


258 


THE  PSYCHOLOGY  OF  NUMBER. 


again,  tlie  quantities  must  be  reduced  to  the  same  unit 
of  measure  :  $6  -r-  Jj^f  =:  ^-^^  -r- 1  =  21  -f-  3  =  8  ;  hence, 
8  yards  can  be  bought. 

Paid  $6  for  f  yard  of  velvet,  what  was  the  price  per 
yard  ? 

The  cost  is  given  in  $6  -i-  f ,  wliich  means  that  J  of 
3  times  the  quantity  sought  is  $6,  and  therefore  it  is 
$6  X  4  -7-  3  =  $8.  Or,  by  the  law  of  commutation,  $8  X 
J  iz.  $f  X  8  =  SO  ;  and  §6  ~  8f  =  88,  as  before. 

(3)  When  both  divisor  and  dividend  are  fractions. 
What  quantity  of  cloth  at  8/75-  a  yard  can  be  bought  for 
8-5-?  The  quantity  is  given  in  8|- -i- 8A,  where  again 
the  quantities  must  be  expressed  in  terms  of  a  common 
unit  of  measure  :  there  results  8J|-  -v-  8-^  =16-7-3  =  5^, 
which  is  the  number  of  yards. 

If  5^  yards  of  calico  cost  8y?  what  is  the  price  per 
yard  ?  AYe  have  8f  -r-  J/ — that  is,  one  third  of  16  times 
some  quantity  =  8f  ;  1^>  times  tlie  quantity  =  8f  X  3  ; 
the  quantity  is  8i  X  3  X  j\  =  8^^. 

The  Inverted  Dimsor. — It  is  obvious  in  all  these 
cases  that  practically  the  divisor  has  been  inverted  and 
then  treated  as  a  factor  with  the  dividend  to  get  the 
quotient.  It  must  be  clear,  too,  that  this  is  simply  re- 
ducing the  quantifies  to  he  compared  to  the  same  imit  of 
measure.  When  8^2  is  to  be  divided  by  Sf — i-  e.,  when 
their  ratio  is  to  be  found — they  must  be  expressed  in  the 
same  unit  of  measure.  The  divisor  is  measured  olf  in 
ffths  of  a  dollar  ;  the  dividend,  then,  must  be  expressed 
in  ffths  of  a  dollar — that  is,  it  becomes  5  X  12,  or  60. 
The  question  is  now  changed  to  one  of  common  division  : 

$12  -^  i  =:  <y> 


4  =  15.     Similarly,  in  StI  ~~ 
$4  —  |o  —  ^1^  ^\^Q  divisor  is  expressed  in  fifths  of  a 


"5" 


f  =  60 


FRACTIONS. 


251) 


dollar;  the  dividend  $||  iiiiist  be  expressed  in  fifths; 
how  is  this  done  ?  By  multiplying  5^} f  l)y  5,  which 
gives  the  number  of  jifih.<  in  ^}  |,  namely,  %\% ;  f<_»r  if 
$12  is  GO  fifths,  ^V  of  $12  must  be  \%  fifths.  The  unit 
of  measure  is  now  the  same,  and  we  have  f  |  (fifths)  -r- 
4  (fifths)  =  1  J.  "  Inverting  the  divisor,"  then,  makes 
the  problem  one  of  ordinary  division  by  expressiiuj  the 
quantities  in  the  same  nmnher  measure. 

Though  formal  proofs  of  rules  are  in  general  too 
abstract  to  begin  with,  yet  after  the  pupil  has  freely 
used  and  learned  the  nature  of  the  processes  involved 
in  concrete  examples,  he  will  quite  readily  comprehend 
the  more  abstract  proof,  and  even  the  general  demon- 
stration.    Take  a  few  instances : 

1.  To  prove  that  the  product  of  two  fractions  has  for 
its  numerator  the  product  of  the  numerators  of  the  given 
fractions,  and  for  its  denominator  the  product  of  their 
denominators : 

(1)  Trove  |  X  |  -  ^^ 


73 


I  X  5  =  ^f- ;  but  this  product  is  9  times 
too  great,  and  therefore  the  required  product  is  \  of 


16 


15 


(2)  f  X  8  rr:  3 ;  for  f  X  S  =  3  X  i  X  8 ;  and 
^  X  9  =  5  ;  multiply  these  equals  ; 

.-.  |X8X|X0  =  3X5;  divide  by  8  X  9 ; 
.  • .  I  X  -|  =  I  Jg  ;    i.  e.,   the    product    of    numera- 
tors, etc. 

(3)  Generally,  let  ^  and  ~  be  any  two  fractions. 

-^  X  ^  =  «  ;  (because,  from  the  nature  of  number, 

-1-  X  ?>  =  1) ;  similarly, 
■J  X  ^  "  <2 ;  multiply  these  equals ; 


i 


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III 


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L   i 


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260  THE  PSYCHOLOGY  OF  NUMBER. 

.".   J  X  ^  Xlcl  =  a  X  c;  divide  these  equals  hyhd; 

•  *  •  T  ^  ^  ~  tv^ '  ^^^^^  ^®'  ^^^^  product  of  tlie  numer- 
ators of  the  given  fractions  is  the  numerator  of  the  re- 
quired fraction,  and  the  product  of  their  denominators 
its  denominator. 

2.  To  prove  the  rule  for  division  of  fractions,  "  in- 
vert the  divisor  and  proceed  as  in  multiplication." 

(1)  f  -i-  4  5  divide  f  by  5  and  there  results  -^-^ ;  but 
it  is  required  to  divide  not  by  5  but  by  ^  of  5 ;  the  re- 
quired quotient  must  therefore  be  9  times  -^^ — that  is, 

a,  which  is  m . 

(2)  |-  X  8  —  3  ;  multiply  these  equals  by  9 ; 
..-.1X8X9  =  3X9.  (1) 

Similarly,  |  X  9  =  5  ;  multiply  these  equals  by  8 ; 
.  • .  I  X  9  X  8  =  5  X  8.  (2) 

Divide  (1)  by  (2) : 

•       3.  _i_  5  —  3x9 

•     •     8     •    "?  ~  5  X  8  • 

Similarly,  a  general  proof  may  be  given,  as  in  mul- 
tiplication. 


h  i' 


by  JcZ; 

lumcr- 

tlie  re- 
inators 


It' 


is,  "  in- 

5? 


the  re- 
tliat  is, 


(1) 

by  8; 
(2) 


D 


mul- 


CIIAPTER  XIY. 


DECIMALS. 


As  already  indicated  in  Chapter  X,  decimals  may 
be  regarded  as  a  natural  and  legitimate  extension  of 
the  notation  with  which  the  pnpils  are  already  familiar. 
Takino;  this  view  of  decimals  as  a  basis  for  teachinir  the 
subject,  we  shall  see  how  easily  and  naturally  all  the 
ordinary  processes  are  established,  and,  further,  liow 
this  mode  of  treatment  recalls  and  confirms  all  that  was 
said  in  building  up  the  simple  rules. 

Notation  and  Numeration. — Consider  the  number 
111 :  the  first  1,  starting  at  the  right,  denotes  one  unit ; 
the  second,  one  ten,  or  ten  units ;  the  third,  one  hun- 
dred, or  ten  tens,  or  one  hundred  units.  The  third  1  is 
equivalent  to  one  hundred  times  the  first  1,  and  to  ten 
times  the  second  1 ;  the  second  1  is  equivalent  to  ten 
times  the  first  1,  and  to  one  tenth  of  the  third  1 ;  the 
first  1  is  equivalent  to  one  tenth  of  the  second  1,  and 
to  one  hundredth  of  the  third  1.  Let  us  now  rewrite 
tlie  number  already  taken,  ])lace  a  point  after  the  first 
1  to  indicate  that  that  1  is  to  be  reii^arded  as  the  unit, 
and  then  place  after  the  point  turee  I's,  so  that  we  have 

111-111. 

AYe  may  ask  what  each  of  these  I's  should  mean,  if  the 

same  relation  is  to  hold  among  successive  digits  that  we 

2G1 


IMAGE  EVALUATION 
TEST  TARGET  (MT-S) 


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33  WEST  MAIN  STREET 

WEBSTER,  N.Y.  14580 

(716)  872-4503 


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202 


THE  PSYCIIULOGY  OF  NUMBER. 


have  supposed  hitherto  to  hold.  The  1  after  the  point, 
standing  next  to  the  1  which  the  point  tells  us  is  to  be 
looked  upon  as  a  unit,  would  naturally  mean  one  tenth 
of  that  1 — that  is,  one  tenth  of  a  unit,  or,  as  we  shall 
say,  one  tenth.  The  next  1,  passing  to  the  right,  stand- 
ing two  places  to  the  right  of  the  unit,  is  one  hundredth 
of  the  unit,  or  one  hundredth ;  it  is  one  tenth  of  the  pre- 
ceding 1 — that  is,  one  tenth  of  one  tenth.  Similarly,  the 
next  1  would  signify  one  thousandth,  and  would  equal 
one  hundredth  of  the  one  tenth  or  one  tenth  of  the  one 
hundredth.  Thus  the  number  above  written  may  be 
read  as  follows :  One  hundred,  one  ten,  one  unit,  one 
tenth,  one  hundredth,  and  one  thousandth.  But  just  as 
in  ordinary  numbers  it  is  convenient,  for  the  purpose  of 
reading,  to  combine  the  elements  into  groups,  here  also 
it  will  be  well  to  adopt  a  similar  method.  The  1  to  the 
extreme  right  is  1  thousandth ;  the  next  1  is,  from  its 
position,  equivalent  to  10  thousandths ;  and  the  next  1 
is  100  thousandths ;  so  that  to  the  right  of  the  point  we 
have  111  thousandths.  The  whole  number  may  now  be 
read,  one  lumdred  and  eleven,  and  one  hundred  and 
eleven  thousandths.  Very  little  practice  will  suffice  to 
acquaint  the  pupil  wdth  the  extended  notation  and  nu- 
meration. A  few  questions,  such  as  the  following,  will 
prove  useful : 

(1)  Read  539*7423,  and  show  that  the  reading  prop- 
erly expresses  the  number. 

(2)  Explain  how  it  is  that  the  insertion  of  a  zero 
between  the  point  and  the  6  in  the  decimal  '5 
changes  the  value  of  the  decimal,  but  that  the  addi- 
tion of  zeros  to  the  right  of  the  5  does  not  change  the 
value. 


DECIMALS. 


263 


ri 


(3)  N^ame  the  decimal  consisting  of  three  digits 
which  lies  nearest  in  value  to  '573245. 

These  will  serve  to  bring  out  in  a  new  relation  some 
of  the  essential  features  of  the  decimal  system,  and  throw 
light  on  some  facts  that  at  an  earlier  stage  in  the  pupil's 
progress  were  necessarily  somewhat  dimly  seen. 

I.  Simple  Rules. 

Multiplication. — When  once  the  notation  is  under- 
stood, addition  and  subtraction  of  decimals  can  oifer  no 
difficulties,  and  we  pass  them  by  to  consider  multiplica- 
tion. In  this  connection  the  most  striking  application 
is  the  multiplication  by  10,  100,  etc.  The  pupil  will  be 
asked  to  compare  7  and  '7,  '3  and  '03,  '009  and  '0009, 
and  he  will  see  at  once  that  the  first  number  in  each 
case  is,  in  virtue  of  the  position  of  the  point,  10  times 
the  second  number.  Next,  when  asked  to  compare  the 
numbers  37  and  3*7,  he  will  see  that  the  3  in  the  first 
number  is  10  times  as  great  as  the  3  in  the  second,  that 
the  7  in  the  first  number  is  10  times  as  great  as  the  7 
in  the  second,  and  that  therefore  the  first  number  is  10 
times  as  great  as  the  second  number.  lie  has  thus  been 
led  to  discover  that  by  moving  the  points  one  place  to 
the  right  we  get  a  immber  10  times  as  great  as  the  origi- 
nal number.  Similarly,  a  corresponding  conclusion  may 
be  reached  for  multiplication  by  100,  1000,  etc.,  and  the 
conclusions  in  each  case  should  be  arrived  at  and  stated 
by  the  pupil.  It  will  at  once  follow  that  to  divide  a 
number  by  10,  100,  1000,  etc.,  we  have  only  to  move 
the  point  one  place,  two  places,  three  places,  etc.,  to 
the  left.  "VVe  pass  next  to  the  multiplication  by  any 
integral  number. 


■I!!] 


iSB 


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■  i[ 
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Hi 


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■'Hi, 


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2G4 


THE  PSYCHOLOGY   OF  NUMBER. 


5-37 
3 


16-11 


4-42 

57 

30-94 

221-0 

251-94 

The  multiplication  in  eacli  of  the  foregoing  cases  is 
based  on  the  same  considerations  as  the  multiplication 
of  integers  by  integers.  Thus,  in  the  second  case,  7 
times  2  hundredths  are  14  hundredths — that  is,  1  tenth 
and  4  hundredths,  and  the  4  must  be  in  the  hundredtlis 
place ;  7  times  4  tenths  are  28  tenths,  which  with  the 
former  1  tenth  make  up  29  tenths  or  2  units  and  9 
tenths,  and  the  9  must  be  in  the  tenths  place ;  thus,  the 
4  and  the  9  will  be  properly  placed  if  the  point  is  in- 
troduced before  the  9,  etc. ;  next,  multiplying  by  5,  we 
must  write  the  results  one  place  to  the  left,  for  reasons 
explained  in  an  earlier  chapter.  The  pupil  will  now 
understand  multiplication  by  an  integer,  and  is  ready 
to  proceed  with  multiplication  by  a  decimal. 

31  2  3-1  2 


23 


2-3 


9-3  6 
6  2-4 
71-7  6 


7-17  6 


He  will  be  asked  to  multiply  some  number,  say  3*12, 
by  some  number,  say  23  ;  the  result  is  71*76.  If,  then, 
we  propose  to  multiply  3'12  by  2-3,  it  will  be  seen  that 
this  differs  from  the  former  only  in  that  the  multiplier 
is  10  times  as  small ;  the  product  then  will  be  10  times 
as  small,  and  may  at  once  be  written  down  7-176.  A 
further  example  or  two,  in  which  a  different  number  of 


DECIMALS. 


2G5 


decimal  places  are  taken,  will  suffice  to  show  that  to 
multii^ly  two  decimals  we  proceed  as  in  the  multiplica- 
tion of  integers,  and  mark  off  in  the  resulting  product 
as  many  places  as  there  are  in  both  multiplier  and  mul- 
tiplicand. 

Division. — To  teach  division,  it  is  well  to  begin  with 
the  division  by  an  integer,  as  this  will  connect  the  pro- 
cess with  what  is  already  known.  Consider  the  follow- 
ing examples : 

(1)  (2) 

T)2 1(3  Y)2-l(-3 

2 1  21 


(3) 
5)-0  0  1  5(-0  0  0  3 
•0015 


(4) 
2  3)1  4  5-8  1(5-4  7 


135 

10-8 
92 


1-61 
1-6  1 


The  pupil  who  can  explain  the  first  division  can  at 
once  explain  the  second,  the  third,  and  the  fourth ;  and 
he  will  see  how  to  divide  whenever  the  divisor  is  a 
whole  number.  Then  he  may  be  asked  to  explain  why, 
in  the  following  divisions,  we  have  the  same  quotient : 

3115  12)60 

5 


5 


He  will  be  led  to  recognise  a  principle  that  he  already 
knows,  namely,  that  the  multiplication  of  divisor  and 
dividend  by  the  same  number  does  not  change  the  quo- 
tient.    He  may  then  be  asked  to  state  a  quotient  equi  va- 


il 


w~ 


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20C 


THE  PSYCHOLOGY  OF  NUMBER. 


I  '■ 
.1 


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\lp..- 


ViUi 


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i ' 


'it  •' ! 


(.,;■ 


lent  to  35  —  "7,  but  having  for  divisor  a  whole  number. 
An  answer  to  be  expected  is,  350  -r-  7 ;  at  any  rate,  he 
can  be  led  to  this  result,  and,  as  this  quotient  is  seen  to 
be  50,  he  can  conclude  that  35  -i-  '7  =  50.  An  exami- 
nation of  a  few  more  examples  will  show  how  always 
to  proceed. 

It  would  be  well  to  have  the  student,  in  his  earlier 
practice,  write  out  a  full  statement  of  what  he  does. 
Suppose  he  is  required  to  lind  the  quotient  1'375'i  -r-  "23 ; 
his  solution  should  stand  somewhat  as  follows : 

The  quotient  of  1"3754  by  "23  is  the  same  as  (multi- 
plying each  number  by  100)  that  of  137*5^1:  by  23. 

2  3)1  3  7-5  •l(5-98 
11  5 

2  2-5 

2  0-7 


1-8  4 
1-8  4 


.-.  1-3754 -^ -23  =  5-98. 

The  delation  of  Decimals  to  Vulgar  Fractions. — 
The  simple  rules  being  understood,  we  may  now  con- 
sider the  conversion  of  decimals  to  the  ordinary  frac- 
tional form,  and  the  conversion  of  ordinary  fractions 
into  the  decimal  form. 

From  the  definition,  '273  =  -f^^  -\-  -j-J^  +  ttott  ~ 
iVo^o,  and  the  student  sees  at  once  how  to  write  a  deci- 
mal in  the  form  of  a  fraction. 

A\^e  may  next  ask  the  pupil  to  divide  1  by  2,  as  an 
exercise  in  the  division  of  decimals. 

2)l-0(-5 
1-0 


DECIMALS. 


2G7 


Similarly, 


4)3-00(-75 
2-8 

20 


8)7-00(-8r5 
6-4 

60 
56 


'« ; 


40 


But  the  quotients  l-f-2,  3-^4,  7-^8  have  up  to 
this  point  been  taken  as  equivalent  to  J,  j,  J. 

,'.  i  =  %i  =  -75, 1  =  -875. 

As  an  exercise  these  results  might  be  verified  thus : 

•^K  —    75    —  3 

A  method  has  now  been  found  for  converting  an  ordi- 
nary fraction  into  a  decimal ;  at  the  same  time  another 
method  has  been  suggested  in  the  verification  above 

made ;  for  we  see  that  4-  =  — ^-  =  — 3^5-^  6_  _  75^  _ 
'  4       2^2      2x5x2x5  —  1^0^  — 

•75.  The  latter  method  is  very  valuable  from  the  point 
of  view  of  theory,  and  the  pupil  should  work  several 
examples  in  this  way. 

We  shall  next  consider  the  example  f . 

3)2-00(-66 
1-8 

20 

18 

2 
.  • .  f  =  '6666  .  .  .,  the  6's  being  repeated  without 
end.  This  fact  is  expressed  thus  :  |  =  '6,  and  f  is  said 
to  give  rise  to  a  recurring  decimal.  Let  us  now  seek 
to  convert  |  to  a  decimal  by  the  other  method.  Accord- 
ing to  it,  we  multiply  the  denominator  by  some  number 
which  will  change  it  into  10, 100, 1000,  etc.— that  is,  into 


n 


« 


i 


p'™ 


11  ; 


Sin: 


li 

J 

! 
t 

'/t 

'l^'T 

,4 
1  ^'1 

J' 

''.1. 

i 

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■:    J;       i 

'1 ! : 

2GS 


THE  PSYCHOLOGY  OF  NUMBER. 


some  power  of  10.  !Now,  any  sucli  power  is  made  by 
multiplying  10  by  itself  some  number  of  times ;  but  10 
itself  is  made  by  multiplying  2  and  5 ;  therefore  every 
power  of  10  is  made  up  wholly  of  the  factors  2  and  5, 
and  in  equal  number.  We  can  not,  then,  multiply  3 
by  any  number  that  will  make  it  into  a  power  of  ten — 
that  is,  we  can  not  convert  f  into  an  ordinary  decimal 
with  a  finite  number  of  digits.  "We  have  thus  a  com- 
plete view  of  the  case.  The  following  are  examples  of 
recurring  decimals : 

•33^  =  -09. 


1^  -  -11:2857 

i--l 

Take  next  ^ : 

6)l-000(-166 
6 

40 

36 

40 

36 
4 

1 


the  6's  recurring,  and  this  is 


•16666  .  . 
written  |^  =  -16. 

Here  the  first  figure  of  the  decimal  does  not  recur, 
and  -J-  is  said  to  give  rise  to  a  mixed  recurring  decimal, 
those  formerly  met  with  being  called  pure  recurring 
decimals.     Similarly, 

■^  =  -583  ^  =  -05. 

If  we  try  to  apply  the  second  method  to  these  examples, 
we  get — 


1  —     1     — 8 5      f  8 

Q  2x3         2x5x3         iF'-'^Tf 

Is  2x2x3 

"i»  2  X  3 ^~3 


2x5x3 

7x5x5 


5x2x5x2x3 
5 


rh  of  -4^ 


6x2x3x3 


=  tV  oi  I 


DECIMALS. 


2G9 


The  examples  given  lead  np  to  tlie  following  proposi- 
tions, for  the  truth  of  which  it  will  be  easy  to  state  the 
general  argument : 

Proposition  I.— A  fraction  whose  denominator  con- 
tains only  the  factors  2  or  5  leads  to  a  decimal,  the  num- 
ber of  w^hose  digits  is  the  same  as  the  number  of  times 
the  factor  2  or  the  factor  5  is  contained  in  the  denomi- 
nator,  according  as  the  former  factor  or  the  latter  occurs 
the  greater  number  of  times. 

Proposition  U.—A  fraction  whose  denominator  con- 
tains  neither  the  factor  2  nor  the  factor  5  leads  to  a 
pure  recurring  decimal. 

Proposition  III.— A  fraction  whose  denominator  con- 
tains in  addition  to  the  factors  2  and  5  a  factor  prime 
to  these  factors,  leads  to  a  mixed  recurring  decimal  the 
number  of  digits  that  are  before  the  period  being'  the 
same  as  the  number  of  times  the  factor  2  or  the  fJictor 
5  is  contained  in  the  denominator,  according  as  the  for- 
mer factor  or  the  latter  occurs  the  greater  number  of 
times. 

Questions  similar  to  the  following  afford  a  valuable 
exercise  on  this  i^art  of  the  work  : 
^      (1)  In  the  case  of  fractions,  such  as  I,  ^V,  etc.,  lead- 
ing to  pure  recurring  decimals,  what  limit  is  there  to 
the  number  of  iigures  in  the  period  ? 

(2)  I  =  -142857 :  explain  why  any  other  fraction 
with  denominator  7  will  lead  to  a  recurring  decimal 
with  a  period  consisting  of  the  same  digits  following 
one  another  in  the  same  circular  order. 

It  will  now  be  in  place  to  consider  the  converse  pro- 
cess of  changing  recurring  decimals  into  their  equiva- 
lent vulgar  fractions.     A  difference  of  opinion  exists  as 


m 


m 


W ' 


1  . 


<!: 


"i;fi'«i 


-Li;, 


i-M^ 


;.;i 


IS'?' 


■  i 


I  -i  ] 


4         i 


270 


THE  PSYCnOLOGY  OF  NUMBER. 


to  the  best  mode  of  dealing  with  these  decimals.  The 
method  here  i)reseiited  is  for  many  reasons  thought  to 
be  the  best : 

•3  =    -3333  .  .  .  (the  3's  repeated  without  end) 

.  • .    -3  X  10  =  3-3333  .  .  .  (the  3's  repeated  without  end) 
and  '3  =    '3333  .  .  .  (the  3's  repeated  without  end) 

.  • .  taking  '3  from  10  times  -3  we  have 

•3X9  =  3 

.•-  •3  =  |  =  i 
which  may  be  tested  by  converting  ^  into  decimal  form. 
Similarly, 

.-.  •i7XlOO  =  17-mnn7  .  .  .  H17  repeated 
•it  =      -17171717  .  .  .  j  without  end) ; 

.  • .  subtracting, 

•17X99  =  17;  .*.  '17  =  ^ 
which  in  its  turn  may  be  tested. 

Next,  to  find  the  fractional  equivalent  of  '279 

-279  =       -2797979  .  .  .  )  /^c)  repeated 

...   -279  X  1000  :=  279-797979     .  .  .  ^  '^[^^J^ 
and  -279  X  10     =     2*797979    ...  J  ^ ' 

.  • .  subtracting, 

-279  X  990  =  279  -  2  =  277 

.     '279-^ 

The  pupil,  after  working  a  few  examples,  will  be  in 
a  position  to  formulate  a  rule  for  writing  down  the  frac- 
tion which  is  equal  to  any  given  recurring  decimal. 

This  treatment  has  the  advantage  of  furnishing  the 
pupil  a  direct  and  definite  method  of  procedure.    Against 


DECIMALS. 


271 


it  is  urged  the  fact  tluat  there  is  made  an  appeal  to  infi- 
nite series,  and  to  the  notion  of  a  limit.  In  reply  to 
this,  it  )nay  be  said  that  in  the  natural  way  of  linding 
the  decimal  which  is  equal  to  a  fraction  we  come  upon 
this  infinite  series — indeed,  we  can  not  avoid  it.  Fur- 
ther, the  notion  of  a  limit — the  term  need  not  be  given 
to  tlie  class— is,  after  all,  not  a  difficult  one  ;  it  may  be 
difficult  to  establish  in  the  case  of  any  given  series  that 
it  has  a  limit,  and  more  difficult  still  to  find  that  Hmit ; 
but  the  difficulty  is  not  in  the  idea  of  limit. 


CONTRACTED   METHODS. 

There  are  few  processes  that  lend  themselves  so 
readily  to  teaching  as  these  contracted  methods.  It 
should  be  explained  to  the  class  that,  very  frequently  in 
physics  and  in  actual  business  life,  operations  have  to  be 
performed  with  decimals,  often  with  many  digits  in  the 
decimal  part,  when  there  are  required  in  the  results 
only  a  few  decimal  ])laces.  For  example,  if  the  answer 
to  a  commercial  problem  were  $79.5017235,  it  would  be 
taken  to  be  $79.59— that  is  to  say,  we  should  take  the 
result  as  far  as  two  places.  In  the  same  way,  if  a  re- 
quired length  were  59-37542156  centimetres,  and  the 
graduated  rule  made  it  imi)ossible  to  consider  anything 
beyond  thousandths  of  a  centimetre,  we  should  take 
the  result  as  59-375.  Accordingly,  if  we  know  in  ad- 
vance that  a  result  of  four  places,  say,  is  all  that  is  re- 
quired, we  may  find  it  possible  to  avoid  unnecessary 
calculations. 

Contracted  MiLltipllcation.  —  Example  :    Multiply 
4-2578532  by  7  correct  to  four  places. 
19 


li 


i 


11 


■I 

lis ' 


'irr 


h 


i         I 

I 
f  . 

I  ' 

.1 

If 

it,  I 

'••  1  ( 


:ii"ii[! 


;ii'' 


1 

,t 

;f*| 

li 

1 

:•(! 

"■■■      , 

1. 


4 


070 


TIIK   PSVCIIOLOCiY   OF  NUMIiEU. 


The  pupil  iiiiiz;lit  be  expected  to  place  the  7,  tlie 
inulti])lier,  beneath  the  8  of  the  iiiultiplieand.  If  he 
does  not  regard  what  is  "carried  "  from  the  multiplica- 
tion of  the  5,  the  complete  multii)lication  might  be  per- 
formed and  the  results  compared  : 


4-25785:32 


29-804i) 


4-2578532 


29-8041)724 


This  one  example  would  serve  to  show  that  wc  must 
take  into  account  what  may  be  carried  into  the  result 
by  the  multiplication  of  the  digit  to  the  right  of  the 
place  directly  considered.  After  the  pupil  has  worked 
a  few  examples  similar  to  this,  he  might  then  be  asked 
to  multiply  4'2578532  by  40  correct  to  four  places.  It 
will  be  seen  that  multiplying  5  (in  the  fifth  place)  by  1 
ten  will  give  5  in  the  fourth  place,  so  that  if  we  place 
4 — which  is  4  tens — beneath  the  5  and  commence  the 
multiplication  there,  we  shall  have  a  result  reaching  to 
four  places,  ^yorking  also  in  full,  the  need  for  taking 
account  of  the  "  carried  "  number  will  appear. 

4-2578532               4*2  5  7  8  5  3  2 
4  40 

17  0-3141  170-3141280 

So,  too,  the  multiplication  of  the  sa?ne  number  by  300 
correct  to  four  places,  will  be  seen  to  be 

4-2578532       4-2578532 
3       300 

12  7  7-3559  1277-3559600 

'Next  take  the  multiplication  of  4*2578532  by  "5,  cor- 


cor- 


DECIMALS. 


273 


roct  to  four  places.  Tlie  7  tlionsandtlis  of  the  multi- 
plicand, if  multiplied  by  1  tentli,  will  give  7  in  the 
fourtli  place,  so  that  we  may  begin  the  multi])lication 
with  the  7. 


4-2578532 


5 


4-2578532 


2-1  2  8  9 


•5 


2-1  2  8  D  2  6  0  0 


If,  now,  we  have  to  multiply  4-2578532  by  347-5  cor- 
rect to  four  places,  all  that  is  necessary  is  to  bring  to- 
gether the  four  results  arrived  at  already : 

4-2578532 
5743 


12  77-35  5  9 
17  0-3141 

2  9-8  0  4  9 
2-1  2  8  9 

1479-()03  8 

But  now  let  us  compare  this  with  the  complete  multi- 
plication : 

4-2578532 
3  4  7-5 


21289 

2  9  8  0  4  9 

1703141 

12773559 


14  7  9-6039 


2660 
724 

28 
6 


8700 


It  will^  be  seen  that  not  all  that  is  necessary  has  been 
taken  into  account,  inasmuch  as  there  is  a  difference  in 
the  results— a  difference  of  1  in  the  fourth  place.     The 


i 


m. 


W 


274 


THE  PSYCHOLOGY  OF  NUMBER. 


f       I 

'  I 


!  ■     , 


I'i    I! 


reason  for  tliis  can  at  once  be  seen  :  tlie  addition  (in  the 
complete  multiplication)  of  the  digits  in  the  fifth  place 
contributes  1  more  than  we  expected  to  be  carried.  To 
guard  against  this  error,  a  certain  precaution  is  taken 
which  may  easily  be  explained.  Suppose  we  are  mul- 
tiplying by  0,  and  we  have  to  multij^ly  7  by  it  to  get 
the  number  to  be  carried ;  this  gives  0  to  carry,  the  3 
belonging  to  the  next  place  and  therefore  being  over- 
looked. Suppose,  in  the  same  multiplication,  w^e  have 
to  multiply  by  6,  and  we  have  to  multiply  8  by  it  to 
get  the  number  to  carry ;  this  gives  4  to  carry,  the  8 
belonging  to  the  next  place  and  therefore  being  over- 
looked. But  already  we  have  overlooked  3  in  that 
place,  making  in  all  11,  which  would  make  a  difference 
of  1  in  the  place  to  which  our  result  is  to  be  correct. 
On  this  account,  when  we  get  such  numbers  as  27,  5G, 
48,  49 — that  is,  with  their  second  digit  greater  than  5 — 
in  the  multiplication  which  is  to  give  the  number  to  be 
carried,  we  consider  them  as  giving  to  carry  3,  6,  5,  5 ; 
whereas  such  numbers  as  32,  42,  54  give  3,  4,  5.  One 
lias  to  use  one's  judgment  about  the  number  to  be  car- 
ried from  such  numbers  as  25,  35,  45.  One  is,  however, 
liable  to  overestimate  or  underestimate,  and  to  secure 
accuracy  to  the  fourth  place,  say,  it  is  generally  best  to 
multiply  to  the  fifth  place. 

Co?itracted  Division. — Suppose  we  have  to  divide 
5-8792314  by  3-421384  correctly  to  3  decimal  places. 
Im^iiediately,  or  by  multiplying  both  numbers  so  as  to 
make  the  divisor  a  whole  number,  we  see  that  the  first 
significant  figure  of  the  quotient  is  in  the  units  place, 
so  that  we  have  to  find  a  quotient  consisting  of  four 
digits,  three  of  which   are   to   the  right  of  the  deci- 


the  8 


DECIMxYLS. 


275 


mal   point, 
formed  : 


First  let   the   ordinary  division   be  per- 


3-421384)5-879'2314(l-7l8 
3421384 


2-457 
2-394 


8474 
9088 


62 
34 


87860 
21384 


28 
27 


664760 
371072 


It  is  plain  that  to  get  the  last  figure  of  the  quotient  we 
needed  only  the  first  two  figures  of  the  third  remainder; 
so  that,  if  a  vertical  line  were  drawn  immediately  to  the 
right  of  these  two  figures,  we  retain  to  the  left  quite 
enough  to  determine  all  the  figures  of  the  quotient,  un- 
less, indeed,  the  subtractions,  etc.,  that  aifect  the  column 
of  figures  to  the  right  of  this  line  may  aifect  the  num- 
bers to  the  left  of  the  line.  It  will  be  noticed  that  to 
the  left  of  the  line  (in  this  case)  in  the  dividend  is  equal 
to  the  number  of  figures  in  our  result.  Let  us  now  try 
to  construct  that  part  of  the  work  which  lies  to  the  left 
of  the  line.     The  first  step  will  stand  thus : 


3-421384)5-879(l 
3-421 

.  2-458 

and  this  means  that  we  consider  only  the  first  four  fig- 
ures of  the  divisor  (there  being  nothing  to  carry  from  the 
multiplication  of  the  fifth).  But  the  remainder  differs 
by  1  in  the  last  place,  which  is  due  to  the  "  carrying  " 
in  the  subtraction  to  the  right  of  the  line  in  the  original 


1% 


h 


I; 


If 


B 


1  I 


'  ) 


,-i-  • 


I;  V'  • 


I 


y 


MR. 


'5i 


276 


THE  PSYCHOLOGY  OF  NUMBER. 


division  ;  however,  let  this  difference  be  overlooked  for 
the  present.  The  next  division,  in  the  figures  to  the 
left  of  the  line,  is  concerned  with  only  three  figures  of 
the  divisor,  there  being  nothing  to  carry  from  the  mul- 
tiplication of  1  by  7 ;  we  shall  now  have  the  work 
thus : 

3-421)5-879(l-7 
3-421 

2-458 
2-394 

64 

Here  the  1  of  the  divisor  w^as  marked  out  after  the  first 
division.  It  will  be  noticed  that  the  remainder  is  here 
64,  while  the  corresponding  number  in  the  complete 
division  is  62.  Referring  to  the  original  work  we  see 
that,  so  far  as  the  figures  to  the  left  of  the  line  are  con- 
cerned, we  have  to  do  only  with  the  first  two  figures  of 
the  divisor.  We  may  therefore  strike  out  the  third 
figure,  and  our  work  will  stand  thus : 

3-4^^)5-879(l-71 
3-421 

2-458 
2-394 

64 
34 

30 

Here  the  remainder  is  30,  while  the  corresponding  re- 
mainder in  the  complete  work  is  28.  Now  strike  out, 
for  the  same  reason  '  as  before  given,  the  4  of  the 
divisor ;  we  are  then  in  doubt  whether  the  last  figure 


DECIMALS. 


277 


'     M 


slionld  be  9  or  8,  as,  taking  9,  we  see  that  9  X  3  =  27, 
wliicli,  with  the  3  to  carry  from  tlie  niulti])lication  of 
4  by  9,  makes  30,  but  we  might  suspect  the  9  to  be  too 
great.  We  see  now  that  if  we  wish  to  have  four  fio-m-es, 
we  should  start  with  a  divisor  of  four  figures.  For  the 
same  reasons  as  given  in  the  case  of  multiplication, 
we  should  also  adopt  a  similar  rule  for  carrying ;  and 
further,  if  we  wish  our  answer  to  consist  of  four  figures, 
we  are  more  likely  to  be  strictly  correct  to  that  pkce  if 
we  start  with  a  divisor  of  five  figures,  which  means  that 
we  retain  an  additional  column  of  figures  of  the  original 
division.     The  work  would  then  stand  thus : 

3-4^1$S)5-8792(l-7l8 
3;4214 

2-4578 
2-3949 

629 
342 

287 
273 

We  have  thus  to  regard  the  following: 

(1)  Find  the  number  of  figures  that  are  to  be  in  the 
answer. 

(2)  Start  the  division  with  a  divisor  consisting  of 
a  number  of  digits  one  more  than  that  number,  these 
digits  to  be  the  first  digits  of  the  given  divisor  in  order. 

(3)  After  each  subtraction,  instead  of  placing  a  fig- 
ure (from  the  dividend)  to  tlie  right  of  the  remainder, 
cut  off  one  figure  from  the  right  of  the  divisor. 

(4)  In  multiplying,  have  regard  always  to  what  may 
be  carried  from  the  neglected  digit  to  the  ri<rht,  re^rard- 


m 


IJ: 


1. 1' " 

I'     I 

I 

I. 

I.'    ' 
.( 


I: 


I'.j:  i,fl   1   . 


278 


THE  PSYCHOLOGY  OF  NUMBER. 


iii<Tr  such  a  number  as  48  as  giving  5  to  carry,  such  a 


number  as  32  as  giving  3. 


Manifestly,  as  in  the  case  of  multipHcation,  there  is 
need  of  practice  to  give  one  confidence,  and  to  educate 
one's  judgment  in  the  matter  of  deciding  what  number 
should  be  carried  in  such  doubtful  cases  as  may  arise. 


■ )  <  -. 

■'')■ 


\k% "! 


M. 


pi ;' 


.'n 


f  > 


-1   ^   \ 


IJ  ;i:i 


CHAPTER  XV. 


1 '« 


ill 


PERCENTAGE   AND   ITS   APPLICATIONS. 

Percentage.— In  some  text-books  on  arithmetic  per- 
centage is  treated  as  if  it  were  a  special  process  involv- 
ing certain  distinctive  principles  and  therefore  entitled 
to  rank  as  a  separate  department.     In  these  books,  ac- 
cordincrlj,  percentage  has  its  definitions,  its  "  cases,"  and 
it?         s  and  formulas.     This  elaborate  treatment  seems 
to  be  a  mistake  on  both  the  theoretical  and  the  practical 
side  :  on  the  theoretical  side,  because  it  asserts  or  assumes 
a  new  phase  in  the  development  of  number ;  on  the  prac- 
tical side,  because  it  substitutes  a  system  of  mechanical 
rules  for  the  intelhgent  application  of  a  few  simple  prin- 
ciples with  which  the  student  is  perfectly  familiar.     In 
the  growth  of  number  as  measurement  percentage  pre- 
sents nothing  new.     It  has  to  do  with  the  ideas  and 
processes  of  ratio  with  which  fractions  are  more  or  less 
explicitly  concerned,  and  its  problems  afford  excellent 
practice  for  enlarging   and  defining  these  ideas,  and 
securing  greater  facility  in  using  them.     But  the  mere 
fact  that,  in  this  new  topic  with  its  cases  and  its  rules,  a 
quantity  is  measured  oif  into  a  hundred  parts  instead 
of  into  any  other  possible  number  of  parts,  appears  to 
be  no  valid  reason  for  constituting  percentage  a  new 
process  marking  a  new  phase  in  the  evolution  of  num- 

279 


'I:ill 


ii 


Sim 


Tf'ra- 


280 


THE  PSYCHOLOGY  OF  NUMBEU. 


I'  » 


I  ■ 


n* ' 


!  ') 


m 


,!  t 


•I 
'"i 


ber.  It  is  no  doubt  correct  enough  to  say  that  "  per- 
centage is  a  process  of  computing  by  hundredths"; 
but  is  such  a  process  to  be  broadly  distinguished  as 
a  mental  operation,  from  a  process  of  computing  by 
eighths,  or  tenths,  or  twentieths,  or  fiftieths?  If  the 
dilference  between  fractions  and  percentage  is  not  a  dif- 
ference in  logical  or  psychological  process,  but  chiefly 
a  difference  in  handlincj  number  symbols,  is  it  worth 
while  to  invest  the  subject  with  an  air  of  mystery,  and 
invent,  for  the  edification  of  the  pu})il,  from  six  to  nine 
"  cases  "  with  their  corresponding  rules  and  f ornmlas  ? 
The  real  facts  regarding  percentage  indicate  clearly 
enough  that,  to  say  tiie  least,  there  is  no  need  for  this 
formal  treatment  and  the  complexity  to  which  it  gives 
rise. 

(1)  The  phrase  ^^^y  cent^  a  shortened  form  of  the 
Latin  per  centum,  is  equivalent  to  the  English  word 
hundredths ;  and  a  rate  per  cent  is,  then,  simply  a 
number  expressing  so  many  hundredths  of  a  quantity. 
Thus,  1  per  cent,  2  per  cent,  3  per  cent,  4  per  cent  .  .  . 
n  per  cent  means  1,  2,  3,  4  ...  ?i  of  the  hundred  equal 
parts  into  which  a  given  quantity  may  be  divided,  ji]st 
^s  tV)  "bV?  t^  '  '  '  Tq  J^cans  1,  2,  3  ...  ^i  of  the  fifty 
parts  into  which  a  quantity  may  be  divided ;  or,  in  gen- 
eral, as  ^i,  "I,  "I,  -^^  .  .  .  represents  1,  2,  3,  4  ...  of 

the  71  parts  into  which  a  quantity  may  be  measured  off. 

(2)  All  problems  in  percentage  involve,  then,  simply 
the  principles  discussed  in  fractions,  and  may  be  solved 
by  direct  application  of  these  principles.  Indeed,  for 
the  mental  w^ork  with  which  every  arithmetical  "7'z<Ze" 
should  he  introduced^  and  by  which  its  study  should  be 
constantly  accompanied,  the  easiest  and  most  effective 


h:^ 


PERCENTAGE  AND  ITS  APPLICATIONS.        281 


treatment  is,  in  general,  by  means  of  the  simplest  forms 
of  fractions  relatively  to  tlie  quantities  involved,  whether 
hundredths,  or  nineteenths,  or  twentieths,  or  ni\\&. 

(3)  All  the  so-called  cases  in  percentage  are  there- 
fore but  direct  applications  of  fractions  as  expressing 
definite  measurement,  or  at  least  may  be  easily  solved 
by  such  applications.  Take  the  following  brief  illus- 
trations of  the  principal  "  cases  "  : 

1.  To  fi7id  any  given  per  cent  of  a  quantity. 

{a)  A  dealer  purchased  a  quantity  of  goods  at  |  of 
their  wholesale  price,  which  was  $325;  what  did  he 
pay  for  the  goods  ? 

{h)  A  dealer  purchased  a  quantity  of  goods  at  CO 
per  cent  of  their  wholesale  price,  which  was  §325 ;  how 
much  did  he  pay  for  the  goods  ? 

Here  {a)  is  a  problem  in  fractions  and  {1>)  one  in 
percentage  ;  in  the  one  case  a  certain  quantity  is  ex- 
pressed as  3-fiftlis  of  another;  in  the  other  it  is  ex- 
pressed as  CO-hundredths  of  it. 

2.  To  find  lohatper  cent  one  quantity  is  of  another, 
{a)  What  part  (fraction)  of  $325  is  $195? 

(h)  What  per  cent  of  $325  is  $195? 

In  a  certain  sense  question  (a)  may  be  said  to  be 
indefinite — i.  e.,  any  one  of  an  unlimited  number  of 
equivalent  fractions  may  be  taken  as  a  correct  answer. 
Thus,  the  answer  is  i||  =  f|  =  f(=:  ^%o^  =  ^%o_  etc.). 
But  if  the  question  were,  how  many  325tlis  of  $325  in 
$195  ?  How  many  65ths  of  it  ?  How  many  fifths  of 
it? — the  respective  answers  to  each  of  these  are  the 
first  three  of  these  fractions,  and  they  are  all  found  by 
exactly  tlie  same  reasoning.  For  example :  1.  -^^j  of 
$325  is  $1 ;  this  is  contained  195  times  in  $195;  tliere- 


(;  li 


^:i . 


m 


m 


I "  '< 


li'' 


■  t     I 


il 


'4.'' ' 


'(  ' 


!  .r( 


li 


h ' 


\J  i 


282 


THE  PSVCIIOLOGY  OF  NUMBER. 


fore,  $195  is  5||  of  $325.  2.  ^V  o^  ^'"^25  is  $5  ;  this  is 
contained  39  times  in  $195;  therefore,  $195  is  §»  of 
$325.  3.  ^  of  $325  is  $65 ;  this  is  contained  3  times 
in  $195 ;  therefore,  $195  is  f  of  $325.  Simihxrly  for 
other  equivalent  fractions  which  answer  corresponding 
questions. 

In  question  (h)  we  are,  strictly  speaking,  limited  to 
one  answer,  but  it  is  found  in  exactly  the  same  way ; 
may,  in  fact,  be  obtained  from  any  of  the  unlimited 
series  of  fractions  that  answer  question  (a). 

The  question  really  is,  how  many  hundredths  of 
$325  are  there  in  $195  ?  We  reason  as  before :  j-J^ 
of  $325  is  $3i  (or  $3.25) ;  this  is  contained  60  times  in 
$195  ;  therefore,  $195  is  j%\  of  $325.  The  solutions  of 
these  questions  might,  of  course,  have  been  varied  by 
Jirst  multiplying  $195  by  1,  65,  5,  and  100  respectively  ; 
thus,  in  question  {h)  the  comparison  is  to  be  made  be- 
tween $195  and  the  hundredth  of  $325 — i.  e.,  how 
often  is  $f^|-  contained  in  $195,  where  (see  Division  of 
Fractions)  the  quantities  must  be  expressed  in  the  same 
unit  of  measure,  and  the  division  is  19500  (hundredths 
of  $1)  -r-  325  (hundredths  of  $1).  In  general,  the  most 
direct  way  is  to  find  any  convenient  fraction  expressing 
the  ratio  of  the  quantities,  and  then  change  this  to  an 
equivalent  fraction  having  100  for  denominator. 

3.  To  find  the  nuinber  of  xoliicli  a  certain  per  cent 
is  given. 

(a)  A  dealer  bought  goods  for  $195,  which  was  ^ 
of  cost ;  find  the  cost. 

(h)  A  dealer  bought  goods  for  $195,  which  was  60 
per  cent  of  cost ;  find  the  cost. 

In  (a)  the  cost  is  measured  off  in  5  equal  parts,  and 


PERCENTAGE  AND   ITS  APPLICATIONS.         283 


1 

TOTT 


3  of  them  are  given :  3  of  them  =  $105,  1  of  them  = 
$05,  5  of  them  (the  whole)  =  $05  X  5  =  $325.  In  {h) 
the  cost  is  conceived  of  as  measured  olf  in  100  equal 
parts,  and  GO  of  them  are  given:  00  of  them  =  $11)5, 
1  of  them  =  $195  -^  00,  100  of  them  (the  whole)  = 
$195  ~  00  X  100  =  $325.  Here,  as  in  the  last  case,  in 
accordance  with  the  principle  connecting  factors  and 
divisors,  we  might  have  nmltiplied  hy  the  respective 
factors  before  dividing  by  the  respective  divisors — e.  g., 
5  times  J  of  a  quantity  =  -J  of  5  times  the  quantity — 
that  is,  $05  ^3X5  ="$05  X  5  -r-  3. 

Introductory  Lesson. — Different  teachers  will  use 
different  devices  in  applying  in  percentage  the  simple 
principles  of  fractions.  The  following  points  are  merely 
suggested : 

1.  It  will  hardly  be  necessai'y,  at  this  stage  of  the 
pupil's  development,  to  use  concrete  illustrations.  It 
will  certainly  not  be  necessary  if  the  pupil  has  been 
taught  arithmetic  according  to  the  psychology  of  the 
subject.  Begin  the  teaching  of  arithmetic  with  the  use 
of  things,  but  do  not  continue  and  end  with  things. 
So  long  as  pupils  have  to  use  objects,  they  are  apt  to 
attend  to  the  mere  practical  processes  at  the  expense  of 
the  higher  mental  processes  through  which  alone  num- 
ber concept  can  arise.  The  infantile  stage  of  depend- 
ence on  objects  is  only  a  stage ;  it  is  not  to  be  a  perma- 
nent resting  place  ;  the  method  of  crawling  on  all-fours 
may  seriously  arrest  development. 

2.  The  first  aim  will  be  to  get  the  pupil  to  identify 
per  cents  with  fractions.  He  already  knows  how  and 
why  a  fraction  may  be  changed  to  an  equivalent  frac- 
tion having  any  given  numerator  or  denominator.     (1) 


i'  n 


1 


ryrp 


I! 
J 


li  I'  * 

'  .  \ 

I  ' 
I . 

I ' 

,1 

1;  • 


!  . 


:!i'l 


I        I 


:M 


:ir 


2S4 


THE   PSYCHOLOGY  OF   NUMIJER. 


Give,  then,  exercises  exprcssiiiir  certain  simple  fractions 
in  (exact  number  of)  Imndredths:  J  =  1%  ;  i  —  i^/a  5 


1    —  _2  0_  .     JL  --     10    .       1 

6  —  10  0?    10  —  T  tnr '    ITU 


5      .      3 
TTJIJ  9    ¥ 


XS 


60    . 


^\  =  ^ijQg.,  etc.  It  will  readily  be  seen  that  a  lar«jje 
number  of  fractions  can  be  changed  into  equivalent 
sliHjle  fractions  liaving  100  for  a  denominator ;  in  other 
M'ords,  into  fractions  expressing  an  exact  number  of 
hundredths.  (2)  Then  some  exercises  to  show  tliat  any 
fraction  may  be  expressed  in  hundredths : 


1  — 


qqi 
100 


>    8 


Ir* .  t  =  ^i .  1 ..  =  ^^  =  'l^'ii  etc 


loo 


100 


2100        100 


The  pupils  already  know  that  multiplication  and  divi- 
sion by  ten  and  by  a  hundred  are  very  easily  performed  ; 
in  other  words,  that  a  number  of  tenths  or  hundredths 
of  a  quantity  is  more  easily  found  than  any  other  frac- 
tion of  that  quantity ;  they  will  also  see  that  the  num- 
ber of  fractions  that  can  be  expressed  as  a  whole  num- 
ber of  hundredths  is  much  larger  than  the  number  that 
can  be  expressed  as  a  whole  number  of  tenths;  they 
will  probably  infer  why  the  ])ractice  of  measuring  off  a 
quantity  in  hundredths  has  been  so  generally  adopted. 

3.  The  different  ways  of  writing  hundredths  will  be 
recalled,  and  the  S3'^mbol  for  the  phrase  2>^^  (^^nt  will  be 
given  ;  for  example,  5  per  cent  has  the  symbol  5^,  and 
is  expressed  by  y-§-^,  5  hundredths,  and  '05. 

4.  Easy  mental  problems  (followed  by  written  work) 
connecting  fractions  with  percentage,  and  illustrating 
the  different  "cases"  of  percentage.  What  fractions 
are  equivalent  to  the  following :  1  per  cent,  10  per  cent, 
25  per  cent,  30  per  cent,  GO  per  cent,  80  per  cent,  90  per 
cent,  etc.  ?     What  per  cent  of  a  quantity  is  -J  of  it,  J  of 


rEJlCKNTACil^]  AND  ITS  APPLICATIONS. 


2S5 


etc. 


off  a 


and 


it,  -J  of  it,  -J-  of  it,  J  of  it,  I  of  it,  J  of  it?  Questions 
like  these,  together  with  practical  i)rol)lems  in  the  same 
line,  will  serve  to  show  the  identity  in  principle  he- 
tween  fractions  and  percentage.  Percentage  is  but 
another  name  for  fractions. 

5.  The  pupils  will  be  then  prepared  for  more  formal 
problems  illustrating  the  general  cases.  These  are  not 
to  be  presented  as  speiyial  cases  demanding  special  rules, 
definitions,  and  formulas.  The  thing  is  to  avoid  mul- 
tiplying rules  and  hair-splitting  definitions,  and  to  give 
the  pupil  facility  in  the  application  of  a  few  simple 
principles.  It  has  been  proved  by  actual  experience 
that  students  who  never  heard  of  the  nine  cases  of  per- 
centage, and  the  nine  rules  or  formnlas,  have  readily 
acquired  the  power  to  handle  any  problem  in  percent- 
age except,  perhaps,  such  as,  on  account  of  their  com- 
])lexity,  are  more  properly  exercises  in  algebraic  analy- 
sis. The  pupil  should  not  be  confined  to  any  one  mode 
of  solution  in  working  problems  in  percentage.  lie  will 
sometimes  use  the  purely  fractional  form,  at  others  the 
so-called  percentage  form,  and  in  still  other  cases  a  com- 
bination of  both  forms.  He  should  be  so  instructed  in 
the  real  nature  of  the  principles  and  practised  in  their 
application  as  to  be  able  to  use  all  forms  with  equal 
facility,  and  almost  instantly  determine,  in  any  given 
problem,  which  of  the  forms  will  lead  to  the  most  con- 
cise and  elegant  solution. 

It  may  be  well  to  utter  a  caution  against  the  vague 
use  of  the  phrase  jf?!?/*  cent^  which  too  generally  prevails. 
It  is  often  used  as  if  it  possessed  in  itself  a  clear  and 
definite  meaning.  It  denotes  simply  a  possible  mode 
of  measurement.     Ten  per  cent,  or  one  hundred  per 


i 


,ii- 


■•',.' 


i;,t 


( 


280 


THE  PSYCHOLOGY   OF  NUMTiKR. 


cent,  lias  no  more  meaning  tlian  ten  or  one ;  all  num- 
bers sigmiy j}Ofi,Hihle  measurements;  tliey  are  empty  of 
meaning  till  applied  to  measured  qua/ititi/.  It  is  not 
U!ieonnnon  to  tind  in  published  solutions  of  percentage 
])roblems  (Itff('re)it  quantities  used  as  defined  by  the  same 
unit  of  measure  because  they  are  expressed  in  per  cents. 
AVe  have  before  us,  for  example,  a  solution  in  which 
the  author  takes  it  for  granted  that  the  difference  be- 
tween 110  per  cent  of  one  quantity  and  DO  per  cent  of 
a  different  quantity  is  20  per  cent.  *'  Let  100  per  cent 
equal  the  recpiired  quantity  "  is  a  very  conunon  pre- 
supposition in  the  solution  of  a  percentage  problem, 
and  equally  common  to  it  to  lind  the  same  100  per  cent 
''  doing  "  duty  for  some  other  quantity  which  demands 
recognition  in  the  same  solution.  So,  in  a  recent  Eng- 
lish work  of  great  pretensions,  we  have  it  posted,  in  all 
the  emphasis  of  black  letter — as  a  fundamental  work- 
ing principle — that  "  100  per  cent  is  1."  One  hundred 
per  cent  of  any  quantity — like  2-halves  of  it,  or  8-thirds 
of  it,  or  4-fourths  of  it  ...  ,  or  n-n\\\%  of  it — is  indeed 
the  quantity  taken  07ice,  or  one  time.  But  this  loose 
way  of  making  "  100  per  cent  equal  to  1,"  or  to  any 
quantity,  is  due  to  a  total  miscon(;eption  of  the  nature 
of  number  as  measurement  of  quantity,  and  of  the  func- 
tion of  the  fraction  as  stating  explicitly  the  process 
of  measurement.  It  seems  as  if  both  teachers  and 
pupils  were  often  hypnotized  by  this  subtle  one  hun- 
dred per  cent. 


Some  ArpLicATiONS  of  Percentage. 

1.  Profit  and  Loss. — We  do  not  need  either  formal 
cases  or  formal  rules,  as  "  given  the  buying  price  and 


i  :■ 


PERCENTAGE   AND   ITS   APITJCATIONS. 


287 


the  selling  price  to  tiiid   the  guiii  or  loss  per  cent." 
A  few  examples  will   serve  to  illustrate  the  different 


''  cases." 


(1)  Bought  sugar  for  C  cents  per  pound  and  sold  it 
for  8  cents  per  pound  ;  lind  the  gain  2)er  cent. 

The  question  simply  stated  is  :  2  cents  gain  on  6 
cents  cost  means  how  much  gain  on  100  cents  of  cost ; 
that  is,  I  =  how  many  hundredths  ?  Multiply  hoth 
terms  by  lOf  (= -Loh)^   '  q,.^ 

0  cents  outlay  ij^ains  2  cents ; 

1  cent       "  "     i  cent ; 

.  • .  100  cents      ''  "     i  X  100  =  3?4  cents. 

(2)  Cloth  was  bought  at  GO  cents  a  yard,  and  sold 
to  gain  25  })er  cent ;  lind  the  selling  price. 

Take  the  cost  price  as  unit  of  comparison  :  Selling 
price  =125  per  cent  of  cost  =  f  cost  =  |  of  60  cents  = 
75  cents.     Or, 

On  100  cents  of  cost  gain  is  25  cents. 
On  1  cent  of  cost  gain  is  -^-^-^  cent. 
.  • .  On  GO  cents  of  cost  gain  is  \  cent  X  GO  =  15 
cents.      Hence   60   cents  -|-  15   cents  =  75   cents,    the 
selling  price. 

(3)  By  selling  cotton  at  12  cents  a  yard  there  is  a 
gain  of  20  per  cent ;  what  was  the  cost  price  ? 

Take  the  cost  price  as  unit  of  comparison  :  20  per 
cent  of  cost  is  \  of  cost ;  therefore,  1^  cost  =  J  cost  = 
12  cents.     Therefore,  cost  =  10  cents. 

(•1)  By  selling  coffee  at  30  cents  a  pound  a  grocer 
lost  25  per  cent ;  what  price  would  bring  him  a  proUt 
of  10  per  cent? 

Selling   price  =  f   of   cost  =  30   cents ;    therefore, 

cost  =  40  cents.     New  price  =  \^  of  cost  =  ^  of  40 
20 


I 


'hi' 


Ml 


h.  .'I 


I! : 


III''!* 


1- 


288 


THE  PSYCHOLOGY  OF  NUMBER. 


cents  =  44  cents.  Otherwise,  the  losing  price,  f  (Jf) 
of  cost,  must  be  increased  to  -J-J^  (|-|)  of  cost — that  is, 
must  be  increased  in  the  ratio  ff ;  therefore,  f|^  of  30 
cents  =  44  cents,  the  price  required. 

(5)  A  merchant  gains  30  per  cent  by  selling  goods 
at  39  cents  a  yard  ;  at  what  selling  price  would  he  lose 
40  per  cent  ? 

Gaining  price  is  |f  of  cost.  Losing  price  is  j\  of 
cost ;  therefore  the  latter  is  y\  of  the  former  =  -^^  of 
39  =  18  (cents). 

2.  Stocl's,  Commission,  etc. — A  few  examples  will 
show  that  there  is  no  new  principle  in  these  rules. 

(1)  How  much  cash  will  be  realized  by  selling  out 
$4,000  stock.  Government  5's,  at  95^  ? 

$100  stock  brings  $95^  cash  ;  $4,000  stock  brings 
$95J  X  40  =  $3,810  cash. 

(2)  What  amount  will  be  realized  by  selling  out 
$4,400  six-per-cent  stock  at  $106f ,  allowing  brokerage  -J-  ? 

Every  100  of  stock  brings  $(106f  -  \)  =  SlOOf ; 
therefore,  $4,400  of  stock  brings  $106J  X  44  =  $4,675. 

(3)  What  semi-annual  income  w'ill  be  derived  from 
investing  $9,000  in  bank  stock  selling  at  $120  and  pay- 
ing 4  per  cent  half  yearly  dividends? 

$120  will  buy  $100  stock,  which  brings  $4  income — 
that  is,  the  income  is  4  -j-  120  =  -^  of  the  investment  = 
^V  of  $9,000  =  $300. 

(4)  Which  is  the  better  investment,  a  stock  paying 
12  per  cent  at  $140,  or  one  paying  9  per  cent  at  $120  ? 
What  income  from  investing  $1,400  in  each  ? 

In  the  first  investment  $140  brings  $12  income ; 
therefore,  $1  brings  %^^-^  =  $-^g-. 

In  the  second  investment  $120  brings  $9 ;  there- 


;[|. 


i 


3    /15\ 

hat  is, 
of  30 


'  fjjoods 
lie  lose 

s  j\  of 
-  -6-  of 

es  will 

s. 

ng  out 

brings 


ng  out 
rage  i  ? 

$1061 ; 
$•1,675. 
)d  irom 
nd  pay- 


come — 
inent  = 

paying 

t  8120^? 

Income ; 
:  tliere- 


PERCENTAGE  AND  ITS  APPLICATIONS.         289 

fore,  II  brings  $^^  =  $^;  $^3^  is  greater  than  $^3_. 
therefore,  the  first  is  the  better  investment. 


Income  from  the  first 


3T 


of  $1,1:00  =  $120 ; 


Income  from  the  second  =  ^\  of  $1,400  =  $105. 

(5)  A  commission  merchant  is  instructed  to  invest 
$945  in  certain  goods,  deducting  his  commission  of  2i 
per  cent  on  the  price  paid  for  the  goods;  find  the 
agent's  commission. 

Since  the  agent  receives  $2^  for  every  $100  he  in- 
vests,  $102i  must  be  sent  for  every  $100  that  is  to  be 
invested  in  goods ;  that  is,  for  every  $102^  sent,  the 
agent  receives  $2J ;  therefore,  he  receives  2^  -j-  102^  = 
jV  of  the  whole  amount  sent;  therefore,  amount  of 
commission  =  $945  -r-  41. 

(6)  For  how  much  must  a  house  worth  $3,900  be  in- 
sured at  2J  per  cent,  so  that  the  owner,  in  case  of  loss,  may 
recover  both  the  value  of  the  house  and  the  premium  paid  ? 

Since  the  premium  is  2^  per  cent  of  the  amount  in- 
sured, the  property  must  be  100  per  cent  —  2^  per  cent  = 
971-  per  cent  =  f »  of  the  amount  insured  ;  therefore. 
If  of  this  amount  =  $3,900,  and  the  amount  is  $4,000. 

(7)  What  amount  nmst  a  town  be  assessed  so  that 
after  allowing  the  collector  2  per  cent  the  net  amount 
realized  may  be  $24,500  ? 

The  collector  gets  2  per  cent  =  j\  of  total  levy ;  there- 
fore, town  gets  11  of  total  levy ;  therefore,  ||  of  total 
hvy  =  $24,500,  and,  therefore,  total  levy  =.  $25,000. 

Interest. 

The  pupil,  having  learned  the  meaning  and  the  use 
of  the  term  per  cent,  should  find  very  little  difticulty  in 
the  subject  of  interest.     However  in  the  problems  of 


I  'j 


'is! 


rfr'rrfi'* 


!  D  = 


!:•(: 


I  » 

H  I 

■ )  ■ 

I :    ' 

.t 


1 


!•;:;' 


J  'I 


i  i  i 


i'- 


200 


THE  PSYCHOLOGY  OF  NUMBER. 


interest  and  kindred  commercial  work  pupils  frequently 
fail ;  but  if  the  cause  of  the  failure  is  examined  into,  it 
will  nearly  always  be  found  to  be,  not  so  much  an  in- 
ability to  meet  the  mathematics  of  the  i)roblems,  as  a 
want  of  accurate  knowled<^e  of  the  terms  used,  and  of 
acquaintance  with  the  business  forms  and  operations 
involved.  On  this  account,  in  taking  up  the  appli- 
cations of  arithmetic  to  commercial  work,  the  teacher 
should  be  at  great  pains  to  ensure  that  every  pupil 
understands  well,  and  sees  clearly  through,  all  such 
forms  and  operations. 

Shnple  Interest. — In  accordance  with  what  has  been 
said,  it  is  necessary  first  to  explain  to  the  class  how  men, 
when  loaning  money,  require  a  certain  payment  for  the 
use  of  the  money,  and  how  the  amount  to  be  paid  for 
this  use — -that  is  to  say,  the  interest — depends  on  the 
time,  twice,  thrice,  etc.,  the  time  (implying,  as  it  does, 
twice,  thrice,  etc.,  the  use),  I'equiring  twice,  thrice,  etc., 
the  interest.  The  unit  of  time  is  generally  taken  as  one 
year,  and  the  7'ate  for  the  year  is  given  as  a  per  cent. 
Accordingly,  if  we  say  that  a  man  loans  money  for  a 
year  at  5  per  cent  per  annum,  we  mean  that  at  the  end 
of  one  year  lie  would  receive  as  interest  -j-lir  ^^  ^^^^ 
money  loaned ;  if  the  money  were  loaned  for  half  a 
year  the  interest  would  l)e  J  of  y^y-  of  the  sum  loaned, 
and  if  for  fifty-three  days,  it  would  be  -^^  of  ^^  of 
the  sum  loaned.  The  pupil  is  now  prepared  to  do  any 
problem  of  calculation  of  simj)le  interest,  and  after  be- 
ing trained  in  the  formal  working  and  stating  of  such 
problems — that  is,  after  realizing  the  problem  to  the 
full — should  be  trained  in  making  rapid  calculations 
after  the  methods  of  men  in  business. 


PERCENTAGE  AND  ITS  APPLICATIONS.         291 


lently 
ito,  it 
an  iii- 
i,  as  a 
Lud  of 
ations 
appli- 
gaclier 
pupil 
[   such 

s  been 
,v  men, 
for  the 
aid  for 
on  the 
it  does, 

as  one 
er  cent. 
y  for  a 
the  end 

of  the 

•  half  a 

loaned, 

T^of 
»  do  any 
ifter  be- 
of  such 
L  to  the 
julations 


He  should  next  be  led  to  see  the  relations  among  the 
interest,  the  sum  loaned  (the  principal),  and  the  sum 
called  the  amount.  Suppose  the  sum  loaned  to  be 
$100,  the  time  to  be  six  months,  and  the  rate  6  per 
cent  per  annum.  Take  the  line  A  B  to  represent  six 
months,  A  the  beginning  of  the  time,  and  B  the  end 
of  the  time. 

$3  interest. 
Principal :  $100       $100  principal. 


A 


B 


At  the  end  of  the  time  the  sum  $103  has  to  be  paid  to 
the  loaner— that  is,  the  $100  has  to  be  restored,  and  $3 
paid  as  interest.  The  sum,  $103,  is  called  the  amount. 
It  is  plain  then  that 

(1)  The  interest    =  -^  of  the  principal ; 

—  T§3  ^^  t^^G  amount. 

(2)  The  principal  =  ^p-  of  the  interest ; 

—  iSt  of  ^^^6  amount. 

(3)  The  amount    =  if2.  of  tlie  interest ; 

—  TW  of  the  principal. 

The  use  of  a  line  to  represent  time  will  assist  the  pupil 
greatly,  and  after  examining  a  few  examples  similar  to 
the  foregoing,  he  will  knoiv  all  the  relations  among  prin- 
cipal, interest,  and  amount,  and  will  see  how  to  write  them 
down  when  the  rate  and  the  time  are  given.  When  these 
relations  are  understood,  tlie  whole  subject  of  interest  is 
understood,  the  only  care  required  on  the  part  of  the 
teacher  consisting  in  making  a  careful  gradation  of 
problems. 

One  of  the  most  striking  applications  of  interest  is 
to  problems  relating  to  the  so-called  true  discount — a 
term  which  should  fall  into  disuse.     There  is  but  one  dis- 


l.ir 


1,1 


ft 


III 


m 


^VfvT^ 


292 


THE  PSYCnOLOGY  OF  NUMBER. 


i 


LI 


W: 


"i- 1 


,r  -' 


count,  the  discount  of  actual  business  life ;  it  is  an  ap- 
plication of  percentage,  and  on  account  of  its  being 
calculated  in  the  same  way  as  interest  it  is  erroneously 
spoken  of  as  interest,  and  a  confusion  arises  in  the  mind 
of  the  pupil. 

Accordingly,  the  problem,  Find  what  sum  would 
pay  now  a  debt  of  $150  due  at  the  end  of  six  months, 
the  rate  of  interest  being  6  per  cent  per  annum,  is  a 
definite  problem  in  interest.  To  solve  it  we  have  re- 
course to  the  line  ilhistration  given  above.  It  is  plain 
that  if  one  had  $100  now,  and  put  it  out  at  ijiterest  at 
the  rate  given,  it  would  come  back  at  the  end  of  the 
time  as  $103.  Thus,  $100  now  is  the  equivalent  of 
$103  at  the  end  of  six  months — that  is,  the  sum  now, 
equivalent  to  a  certain  sum  due  at  the  end  of  six 
months,  is  \^^  of  that  sum.  Therefore,  in  the  case  in 
question,  the  sum  is  UJ  of  $150.  It  is  true  that  there 
is  here  an  allowance  off,  a  discount,  so  to  speak,  but 
until  the  pupil  understands  the  whole  question  of  inter- 
est and  discount  the  term  should  not  be  used  in  this 
connection.  We  shall  suppose,  then,  that  the  student 
has  mastered  simple  interest,  and  shall  turn  to  com- 
pound interest. 

Com/pound  Interest. — The  teacher  should  explain 
that  the  value  of  money — as  the  pupil  has  seen — de- 
pends, in  some  measure,  on  where  it  is  placed  in  time ; 
men  in  business  always  suppose  interest  to  be  paid  when 
it  is  due,  or  if  an  agreement  is  made  that  its  payment  be 
deferred,  they  regard  this  interest  in  its  turn  a  source  of 
interest.  An  example  worked  out  in  detail  will  help 
the  pupil  to  see  just  what  is  done.  Suppose  a  sum  of 
,000  loaned  for  three  years,  at  5  per  cent  per  annum, 


PERCENTAGE  AND  ITS  APPLICATIONS.         293 


interest  to  be  paid  at  the  end  of  the  three  years,  and 
the  interest  at  the  end  of  each  year  to  become  a  source 
of  interest  for  the  ensuing  year  or  years : 

$10  0  0  Principal. 
5 

$5  0.0  0  First  year's  interest. 
$10  0  0 

Sum  bearing  interest  for  the  2d  year. 


$105  0 
5 

$5  2.5  0 
$10  5  0. 

$110  2.5  0 
5 


$5  5.1  2  5  0 
1 1  0  2.5  0 

$1 1  5  7.6  2  5  0 
1  0  0  0.0  0 

$1  5  7.6  2  5 


Second  year's  interest. 

Sum  bearing  interest  for  the  3d  year. 

Third  year's  interest. 

Amount  to  be  paid  at  end  of  time. 
Original  principal. 

Amount  of  interest. 


The  pupil  will  work  several  such  examplefi,  and  will 
find  not  a  little  pleasure  in  determining  just  how  much 
interest  has  been  paid  as  interest  on  interest.  He  is 
then  ready  to  make  a  more  general  study  of  compound 
interest. 

Suppose  a  sum  loaned  at  compound  interest  for  three 
years  at  5  per  cent  per  annum.  What  is  the  interest  on 
any  sum  for  one  year  at  5  per  cent  ?  Plainly  yj^  of  the 
sum.  What  is  the  amount  ?  -fH-  of  the  sum.  What, 
then,  is  the  amount  of  any  sum  for  one  year?  -ffj 
of  that  sum.  What  sum  bears  interest  for  the  second 
year  ?    fJJ  of  the  original  sum.     What  will  the  amount 


I  !   ,, 


%m 


■  ^^  II 

.it 


^H'fTff'^ 


294 


THE  PSYCHOLOGY  OP  NUMBER. 


ii'i 

lit  r?- 


Mi'  i 


III 


,  ll. 


■f. 


'If 


of  this  be?  |J|  of  itself,  and  therefore  {^  of  fg J  of 
the  original  sum.  Accordingly,  the  amount  of  the  sum 
for  two  years  is  (xg-f)'  of  the  original  sum.  What  for 
three  years?  Plainly  -}-§|-  of  (y§§-)'  of  original  sum, 
and  therefore  ({-§■§/  of  original  sum.  This  is  found 
to  be  "HfJ^I-J-  of  original  sum.  How  much  more 
have  we  than  the  original  sum  ?  tVoVWjt  ^^  original 
sum  ;  therefore,  interest  —  iVVoVo^o  ^^  sum  —  xVVtuVs" 
of  amount,  etc. 

The  pupil  should  be  told  that  in  all  transactions  in- 
volving a  time  longer  than  one  year  (or  it  may  be  by 
agreement  six  months  or  three  months)  compound  in- 
terest is  alone  employed  where  tlie  interest  is  thought 
of  as  all  being  paid  at  the  end  of  the  time.  From  what 
has  been  said  he  will  know  at  once  how  to  solve  the 
following  problem  of  interest :  Find  what  sum  paid 
now  will  discharge  a  debt  of  $1,000,  due  at  the  end  of 
three  years,  the  7'ate  of  interest  being  6  per  cent.  He 
should  acquire  a  facility  in  thus  transferring  money 
from  one  time  to  another. 

Annuities. — Afew  words  may  be  said  on  the  sub- 
ject of  annuities.  If  A  gives  B  $100  to  keep  for  all 
time,  and  the  rate  of  interest  be  6  per  cent,  B  would  be 
undertaking  an  equivalent  if  he  would  agree  (for  him- 
self and  his  heirs)  to  pay  to  B  (and  his  heirs)  $6  at  the 
end  of  each  year,  for  all  time.  This  $0  supposed  paid  at 
the  end  of  each  year  is  called  an  annuity  ;  as  it  runs  for 
all  time,  it  is  called  a  perpetual  annuity,  and  is  said  to 
begin  now,  though  the  first  payment  is  made  at  the  end 
of  the  first  year.  The  $100  is  very  properly  called  its 
cash  value,  and  the  relation  of  the  $100  to  the  annuity 
of  $6  is  plainly  that  of  principal  to  interest.     Thus,  it 


WW 


lly:; 


3  sum 
at  for 

sum, 
found 

more 
•iffinal 

)7635 

ns  in- 
be  by 
nd  in- 
lousjbt 
1  wbat 
^e  tbe 
1  paid 
end  of 
t.  He 
money 

le  sub- 
for  all 
Duld  be 
)r  him- 
)  at  tlie 
paid  at 
uns  for 
said  to 
tbe  end 
died  its 
annuity 
Thus,  it 


PERCENTx\GE  AND  ITS  APPLICATIONS. 


295 


will  be  easy  to  find  tbe  cash  value  of  any  given  per- 
petual annuity,  or  to  find  the  perpetual  annuity  that 
could  be  purchased  with  a  given  sum.  To  illustrate 
this  we  should  need  a  line  extending  beyond  all  limits : 


$100    $()      $6      $()      $()      $0 


$0      $6 


> 


(The  divisions  of  the  line  represent  each  one  year.) 
Kext  we  may  su})pose  an  annuity  to  begin  at  the 
end  of,  say,  three  years,  so  that  the  first  payment  would 
be  made  at  the  end  of  the  fourth  year.  Taking  the 
annuity  to  be  $6  and  the  rate  6  per  cent  per  annum, 
we  see  that  the  value  of  this  annuity  at  the  beginning 
of  the  fourth  year  (represented  by  the  point  in  the  illus- 
tration below)  is  $100. 


$100    $0      $0      $6 


I 


B       C       D      E       F 


J 


H 


> 


But  that  $100  is  placed  at  the  end  of  three  years  from 
now,  and  is  therefore  equivalent  to  (fg^)'  ^^  ^^^^  "^^^'• 
We  have  thus  the  cash  value  of  an  annuity  deferred 
three  years. 

When  the  pupil  knows  how  to  deal  with  the  two 
cases  discussed  he  can  easily  be  led  to  find  the  cash 
value  of  an  annuity  beginning  now  and  running  for  a 
definite  number  of  years.  When  asked  to  compare  the 
two  perpetual  annuities  represented  below,  he  will  see 
that  the  first  exceeds  the  second  by  three  payments — $6 
at  the  end  of  the  first  year,  $0  at  the  end  of  the  sec- 
ond year,  and  $0  at  the  end  of  the  third  year,  and 
these  constitute  an  annuity  for  three  years  beginning 
now. 


\\ 


M 


■f|Tr— 

w    ! 

m  :■ 

i.-,r'  . 

I ;  ;  '  i 


I, 


:  ;  I    ,  ■ 


296 


THE  PSYCHOLOGY  OF  NUMBER. 


$6  $6  $6  $0  $0  $()  86  $6 
1    1    1    1    1    1    1    1 

$6 
1 

$6  $6  $6  $6  $6 
1    1    1    1    1    1    1    1 

$6 
1 

> 


> 


But  the  cash  vahie  of  the  first  annuity  is  $100,  and  the 
cash  value  of  the  second  is  (|^J)'  of  $100. 

.  • .  The  cash  vahie  of  an  annuity  of  $6  beginning 
now,  running  for  three  years,  is  $100  —  (-}^§)'  of  $100 
or  U -(«!/}  of  $100. 

It  will  be  easy  to  obtain  a  general  formula,  and  also 
to  find  the  value  of  a  deferred  annuity  running  for  a 
definite  number  of  years. 


m 


ii  ■■if' 

i 


:  f 
:'i 

■  f 

'■ri 


t     ! 


CHAPTER  XYL 


EVOLUTION. 

Square  Root. — The  product  of  3  and  3  is  9 ;  of  5 
and  5  is  25.  The  measures  of  squares  whose  sides 
measure  3  and  5  are  9  and  25.  We  say  that  9  is  the 
square  of  3,  and  that  25  is  the  square  of  5 ;  3  is  the 
square  root  of  9,  and  5  the  square  root  of  25.  The 
square  of  3  is  written  3",  the  square  root  of  3  is  ex- 
pressed thus :  V^d.  The  pupil  can  write  at  once  the 
table  of  squares : 

1'=    1 

2»=    4 

3'=    9 

4"  =  16 

5'  =  25  [10'  =  100] 

6'  ==  36 

7^  =  49 

8'  =  64 

9"  =  81 

He  will  note  that  the  square  of  any  number  expressed 
by  one  digit  is  a  number  expressed  by  one  digit  or  by 
two  digits,  while  the  lowest  number  expressed  by  two 
digits — viz.,  10 — has  for  its  square  100,  a  number  ex- 
pressed by  three  digits. 

297 


!-i 


1"^? 


w: 

f 

i 

1 

'i 

1 

1 

1: 

1 

wm 


It  .(•  ' 

!:;■'•■ 


In: 


lit : 


■  i  K 


f 

1 
,  ^j  4       -      : 

■J.  t       > 

ill  .   ^ 

m     \        ^.        ; 

I'll 

:  i;jL 

^ 

■  ttiiiL 

298 


THE  PSYCHOLOGY  OF  NUMBER. 


It  is  plain  that  the  square  of  any  number  expressed 
by  two  digits  has  for  its  square  a  number  expressed  by 
three  digits  or  by  four  digits.  Also  the  square  root  of 
a  number*  expressed  by  three  digits  is  a  number  ex- 
pressed by  two  digits,  and  the  tens  digit  is  known  from 
the  first  digit  on  the  left ;  for  example,  025  (if  it  has 
an  exact  square  root),  lying  as  it  does  between  400  and 
900,  will  have  for  square  root  a  mimber  lying  between 
20  and  30 — that  is,  the  tens  figure  of  the  root  will  be  2. 
Similarly,  if  a  number  is  expressed  b}^  four  digits  its 
square  root  is  expressed  by  two  digits,  and  the  tens 
digit  of  the  root  can  be  determined  from  the  first  two 
digits  (to  the  left)  of  the  number ;  thus  the  square  root 
of  2709 — a  number  lying  between  2500  and  3600 — will 
have  5  for  a  tens  digit,  and  this  is  determined  by  the 
27  of  the  number  2709. 

AVrite  next  the  table  of  squares ; 


10' 

20" 
30" 
40' 
50' 
60' 
70' 
80' 
90' 


100 
400 
900 
1600 
2500 
3600 
4900 
6400 
8100 


[100'  =  10000] 


Now  take  13  and  square  it :  the  result  is  169.     We 
wish  to  arrive  at  a  method  of  recovering  13  from  169. 


*  In  general,  when  we  speak  of  the  square  root  of  a  number,  wo 
suppose  that  it  has  an  exact  square  root. 


EVOLUTION. 


299 


To  do  tliis  we  shall  examine  how  the  1G9  is  formed 
from  the  13 : 

13 

13  • 

9 

30) 
30  f 
100 

169 

Thus  13,  which  is  made  up  of  two  parts,  10  and  3,  has 

for  its  square  a  number  169,  which  is  seen  to  be  made 

up  of  100,  the  square  of  10 ;  9,  the  square  of  3 ;  and 

twice  the  product  of  10  and  3.     This  is  familiar  to  the 

pupil  who  has  worked  algebra,  and  may  be  ilhistra- 

ted    geometrically. 

The  whole   square  ^^ 

is  measured  by  13", 

and  its  parts  make 

up   10'  +  2  X  (10 

X  3)  +  3'. 

Now,  to  recov- 
er 13  from  169 : 
we  see  that  its 
hundreds  digit  1, 
showing  that  the 
number  lies  be- 
tween 100  and  400, 
gives' the  tens  digit 

of  the  root,  so  that  we  know  one  of  the  parts  of  the 
root,  viz.,  10.  The  square  of  this  part  is  100,  and  the 
rest  of  the  given  number,  69,  must   be  2  times  10, 


III 


8 


ipji' 


?'! 


IT?: 


'(■ ' 


.•'If 


'*i 


300 


THE  PSYCHOLOGY  OP   NUMBER. 


multiplied    by  the  other  part,  together  with  the  square 

of  the  other  part. 

1GO|10 

lUO 

69 

Accordingly,  if  we  multiply  10  by  2  and  divide  09  by 

this  product  we  get  a  clue  to  the  other  part.    Dividing 

60  by  2  X  10,  or  20,  we  see  that  the  quotient  is  a  little 

greater  than  3  ;  if,  then,  after  taking  3  times  20  from  09 

there  is  left  the  square  of  3,  we  have  the  root.     Plainly 

this  is  the  case  : ' 

100|10  +  3 

100 


20 


01> 
CO 


9  =  3' 

Now,  this  work  might  be  written  somewhat  more  neatly, 

thus: 

109|10  +  3 


100 

20  ) 
+  3) 

09 

09 

It  may  be  further  simplified  by  leaving  out  unneces 

sary  zeros,  thus : 

10913 

1 

23 

09 

69 

The  pupil  is  now  in  a  position  to  find  the  square 
root  of  all  numbers  expressed  by  three  or  four  digits. 
It  would  be  well,  before  considering  the  squaro  root  of 


li  ! 


EVOLUTION. 


301 


larger  mimbcrs,  to  exaniiiio  for  the  square  root  of  sueli 
numbers  as  IMJO,  27'09.  The  pupil  will  see  at  ouco 
that  the  scpuire  root  of  the  former  number  lies  between 
1  and  2,  that  of  the  latter  between  5  and  0,  and  can 
easily  be  led  to  complete  the  process  of  extracting  tlie 
roots,  iinding  as  results  1*3  and  5*3.  He  will  thus  dis- 
cover for  himself  that  the  problem  is  not  different  from 
the  one  already  solved. 

We  are  now  ready  to  examine  for  the  square  root 
of  larger  numbers.     Write  first  the  following  table  : 

100"=  10000 

200'=  40000 

300'=  90000 

400'  =  160000 

500'  =  250000     [1000'  =  1000000] 

600'  =  360000 

TOO'  =  490000 

800'  =  640000 

900'  =  810000 

A  study  of  the  table  will  lead  to  the  conclusion  that 
the  square  roots  of  numbers  expressed  by  5  or  6  digits 
are  numbers  expressed  by  3  digits,  and  that,  if  expressed 
by  5  digits,  the  first  (to  the  left)  digit  of  the  root  is 
determined  by  the  first  (to  the  left)  figure  of  the  num- 
ber, and  if  expressed  by  6  digits,  by  the  first  two  digits 
of  the  number.  Thus,  the  square  root  of  16900  will 
have  1  as  the  hundreds  digit,  while  that  of  270900  will 
have  5  as  the  hundreds  digit.    Next,  by  multiplication, 

we  find  that 

230'  =  52900 

and  240'  =  57600 


'I 


iii 


802 


THE  PSYCnOLOGY  OF   NUMBER. 


I'       / 

I    ; 
I 

,1 


1 


Si 


I  1  .    1    c 

m 


'  i- 


!i 


1  !i 


lii 


Conserjnciitly  the  square  root  of  (say)  5475G  must  lie  be- 
tween 230  and  240 — that  is,  mns't  have  23  as  its  first  two 
digits.  The  first  tliree  digits  of  the  number  54756  are 
sufficient  to  show  that  this  must  be  the  case.  Kow,  sup- 
pose we  seek  the  square  root  of  54750. 

Phiinlvj  the  first  part  of  the  root  is  200 : 

5475G|200 

40000 

"14750 

Now,  had  we  been  seeking  the  square  root  of  52900, 
wliich  is  230 — that  is,  consists  of  two  parts,  200  and 
30 — we  should  have  worked  thus : 


529001200  +  30 
40000 

200  X  2  -  400  ) 
30  S 

12900 
12900 

for  57600  : 

57600  200  +  40 
40000 

200  X  2  -  400  ) 
+    40  i 

17600 
17600 

Then  plainly  we  see  how,  in  finding  the  square  root  of 
54756,  to  determine  the  second  figure : 


'?i  I 


200  X  2  =  400  I 
+  30  i 


547561200  +  30 
40000 
14756 
12900 


1856 


EVOLUTION. 


303 


;  lie  be- 
rst  two 
T56  are 
i\v,  siip- 


52900, 
>00  and 


3  root  of 


We  have  yet  to  find  the  units  digit  of  the  root.  But 
at  tins  point  we  may  say  that  the  root  consists  of  two 
parts,  one  230,  and  the  other  to  be  foun^  and  may  pro- 
ceed as  in  the  earlier  case  : 


400  I 

430 
230  X  2  =  460  I 

464 


547561200  +  30  +  4 

4000J) 

1475^ 

12900 


1856 


1856 


The  work  may  now  be  shortened : 

54756|234 


43 

464 


147 
129 


1856 
1856 


After  the  pupil  has  been  exercised  in  extracting  the 
roots  of  numbers  expressed  by  5  or  6  digits,  he  will  find 
no  difficulty  in  determining  the  square  roots  of  such 
numbers  as  547-56,  5-4756,  '054756.  The  extension  to 
numbers  expressed  by  a  higher  number  of  digits  will 
be  easy,  and  the  need  for  marking  off  into  periods 
of  two,  starting  from  the  decimal  point,  as  well  as 
its  full  significance,  will  have   been  realized  by  the 

^""^Up  to  this  point  we  have  spoken  of  numbers  whose 
21 


myi'r 


I  « 


A.: 


m 


'>    »! 


\<Tt' 


i!M-' 


'(■■! 

1  ■ 


ll'l 


W'' 


''l 


I 


Bi 


I  ^ 


304 


THE  PSYCHOLOGY  OF  NUMBER. 


square  root  can  be  extracted ;  it  will  be  next  in  order 
to  deal  with  the  approximations  to  square  roots— for 
example,  the  square  root  of  2,  5,  etc. ;  but  as  this  in- 
volves nothing  essentially  new  it  will  not  be  here 
discussed. 

"We  shall  conclude  this  part  of  the  work  by  calling 
attention  to  the  extraction  of  the  square  root  of  a  frac- 
tion.   Since 

3      3      9' 

the  square  root  of      -  =  — =  =  -  • 
^  9^/9       3 

In  the  case  of  fractions  whose  denominators  are  num- 
bers whose  roots  can  not  be  exactly  determined,  we 
should  proceed  as  follows : 


Take,  for  example, 


A- 


'^  3  3X3         3' 

an  artifice  the  value  of  which  is  apparent. 

Cube  Hoot. — The  method  of  teaching  square  root 
has  been  presented  in  such  detail  that  very  few  w^ords 
will  suffice  on  the  subject  of  cube  root. 

From  examples  such  as  3'  =  27  the  meaning  of 
cube  and  cube  root  will  be  brought  out,  and  use  may 
be  made  of  the  geometrical  illustration  of  the  cube. 
The  pupil  should  commit  to  memory  the  following 
table : 


t 


EVOLUTION. 


305 


order 
— for 
is  in- 
here 

ailing 
,  f  rac- 


Tium- 
id,  we 


re  root 
words 

ling  of 
se  may 
B  cube, 
llowing 


[10'  =  1000] 


V=     1 

2"=  8 
3'=  27 
4'z=  64 
5'  =  125 
6'  =  216 
7' =  343 
8'  =  512 
9'  =  729 

All  numbers  expressed  by  1,  2,  or  3  digits  have  for 
cube  roots  numbers  expressed  by  1  digit. 
Next  we  have  the  following  table  : 

10'  =  1000 
20'  =  8000 
30'=  27000 
40'=  64000 
50' =  125000 
60'  =  216000 
70' =  343000 
80'  =  512000 
90*  =  729000 

Thus  all  numbers  expressed  by  4,  5,  or  6  digits  have 
for  cube  roots  numbers  expressed  by  2  digits ;  further, 
the  first  digit  of  the  root  in  such  case  is  determined  by 
the  first  one  (to  the  left),  the  first  two,  or  the  first  three 
digits  of  the  number,  according  as  it  is  expressed  by  4, 
5,  or  6  digits.  Thus  the  cube  roots  of  the  numbers 
2744,  39304,  357911,  will  in  every  case  be  numbers  ex- 
pressible by  two  digits.  The  tens  digits  will  be  deter- 
mined by  the  2,  the  39,  the  357  of  the  numbers  to  be 
1,  3,  7  respectively. 


[100'  =  1000000] 


\m  f 


]!» 


I 


t  ■ 


li  '  If 


■:i  ■■ 


11^ 


i 


n'' 


,t<|S' 


;'ir 


I 


i'^^; 


till 


306 


THE  PSYCHOLOGY   OF  NUMBER. 


To  lind  the  cube  root  of  a  niiinber  we  shall  see  how 
the  cube  of  a  number  is  formed.  The  identity  of  the 
following  two  ways  of  multiplying  14  by  14,  and  the 
product  by  14,  will  at  once  be  seen : 

10+4 
10+4 


10^ 
10' 


(4  X  10)  +  4» 
+    (4  X  10) 
+  2(4  Xl0)  +  4» 

10+4 


(4Xl0')  +  2(4»Xl0)  +  4'' 
10«  +  2  (4  X  10')  +    (4'XIO) 


14 
14 

56 
14 

196 
14 

784 
196_ 
2744 


10'  +  3  (4  X  10')  +  3  (4'  X  10)  +  4' 

We  see,  then,  that  the  cube  of  such  a  number  as  14 — 
that  is,  a  number  regarded  as  being  made  up  of  two 
parts,  here  10  and  4 — is  the  cube  of  one  part  increased 
by  three  times  the  product  of  the  square  of  that  part 
and  the  other  part,  and  three  times  the  product  of  the 
first  part  and  the  square  of  the  second,  and  the  cube  of 
the  second  part. 

We  wish  now  to  recover  from  2744  its  cube  root. 
Plainly,  the  tens  digit  of  the  root  is  1 — that  is,  the  first 
part  of  the  root  is  ten ;  take  from  2744  the  cube  of  10 : 

2744|10 

1000 

1744 

The  remainder  is  made  up  of  three  parts : 

(1)  The  product  of  3  times  the  square  of  10,  and  the 
other  part  of  the  root. 

(2)  The  product  of  3  times  10,  and  the  square  of 
the  other  part  of  the  root. 


EVOLUTION. 


307 


(3)  The  cube  of  the  other  part  of  the  root. 

Then,  if  \\e  divide  1744  by  3  times  the  square  of 
10,  we  sliall  have  a  clue  to  the  other  part  of  the  root. 
Dividing,  we  may  take  5  as  the  other  part : 


3  X  10'  =  300 


27441 10 +  5 
1000 
1744 
1500 


244 


Taking  away  3  X  10'  X  5,  we  have  as  remainder  244, 
which  should  be  made  up  of  parts  (2)  and  (3),  men- 
tioned above.  We  find,  however,  that  it  is  not  large 
enough.  We  have  taken  a  second  part  too  large,  and 
therefore  take  a  smaller  part,  say  4 : 


2744|10  +  4 
1000 


3  X  10'  =  300 


1744 
1200 


544 


Here  the  remainder  is  544,  which  =  3  X  10  X  4'  +  4', 
and  we  conclude  that  the  root  is  14. 

We   see   also  that   the   1744  =  4(3  X  6'  +  4  X  3  X 
10  +  4').     The  work  might  then  be  shown  thus : 


274410  +  4 

1000 

3  X  10'  -  300  ) 

1744 

4  X  3  X  10  -  120  >• 

4'  -  If,  ) 

436 

1744 

-/.'V' 


ll 


I'i  .'Ik     1  ■ 


308 


THE  PSYCHOLOGY   OF  NUMBER. 


I;    ^i!   «i   B 


"II, 


I: 


4;i 


■|1. 


,1, 

''1 

i 

i         1 
1,1     '   ' 

■  1 

1    1 

1 

m. 

It  may  be  further  sliorteiied,  thus : 


2744|14 
1 


1'  X  300  =  300 

1  X  4  X  30  =  120 

•       ^'^    16 

436 


1Y44 


1744 


The  further  development  of  the  method  will  follow 
lines  similar  to  those  followed  in  square  root.  We  shall 
take  spLr  .  Li  .',y  to  indicate  how,  in  the  case  of  finding  a 
cube  root  cohsi  ling  of  several  figures,  a  certain  saving 
of  work  may  be  s^c^^red  ; 

814  780  504|934 


729 


OO^'XS 
3  X  90  X  3 
3' 


24300 


=      810^ 

= 9 

3  X  90'+ 3  X  (3  X  90)4-3'=  25119  ^ 

9 

2594700 
11160 

16 

2605876 


85  780 


75  357 


10  423  504 


10  423  504 


When  ve  reach  the  point  where  we  wish  to  determine 
the  third  figure,  we  have  to  find  three  times  the  square 
of  93 — that  is, 

3(90*+2X90X  3  +  3') 

Kow,  as  indicated  above,  810  is  3  X  90  X  3,  9  is  3', 
25119  is  3  X  90'  + 3  X  90  X  3  +  3',  so  that,  if  to  the 
sum  of  810,  9,  25119  we  add  9,  which  is  the  square  of 


EVOLUTION. 


309 


How 
;hall 

nng 


3,  we  shall  have  found  three  times  the  square  of  93. 
If  to  the  resulting  number  we  affix  two  zeros,  we  shall 
have  three  hundred  times  the  square  of  93. 

This  artifice  may  be  employed  when  at  each  suc- 
cessive stage  we  need  three  hundred  times  the  square 
of  the  part  already  found. 

We  shall  conclude  this  chapter  with  the  remark  that 
the  fourth  root  of  a  number  is  to  be  found  by  extracting 
the  square  root  of  its  square  root,  and  the  sixth  root  of 
a  number  by  extracting  the  square  root  of  its  cube  root. 


|934 


line 
lare 


THE    END. 


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D.   APPLETON  &   CO.'S  PUBLICATIONS. 

"  No  library  of  military  literature  tlint  has  appeared  in  recent  years  has  been  so  in- 
structive to  readers  of  all  kinds  as  the  Great  Commanders  Series,  which  is  edited  by 
General  James  Grant  Wilson." — New  York  Mail  and  Express. 


r^REAT    COMMANDERS,       A   Series  of    Brief 

^^     Biographies   of    Illustrious    Americans.       Edited   by   General 

James  Grant  Wilson.    i2mo,  cloth,  gilt  top,  $1.50  per  volume. 

This  series  forms  one  of  the  most  notable  collections  of  books  that  has 
been  published  for  many  years.  The  success  it  has  met  with  since  the  first 
volume  was  issued,  and  the  widespread  attention  it  has  attracted,  indicate  that 
it  has  satisfactorily  fulfilled  its  purpose,  viz.,  to  provide  in  a  popular  form  and 
moderate  compass  the  records  of  the  lives  of  men  who  have  been  conspicu- 
ously eminent  in  the  great  conflicts  that  established  American  independence 
and  maintained  our  national  integrity  and  unity.  Each  biography  has  been 
written  by  an  author  especially  well  qualified  for  the  task,  and  the  result  is 
not  only  a  series  of  fascinating  stories  of  the  lives  and  deeds  of  great  men, 
but  a  rich  mine  of  valuable  information  for  the  student  of  American  history 
and  biography. 

The  volumes  of  this  series  thus  far  issued,  all  of  which  have  received  the 
highest  commendation  from  authoritative  journals,  are  : 

ADMIRAL   FARRAGUT.     By  Captain  A.  T.  Mahan,  U.  S.  N. 

GENERAL   TAYLOR.      By  General  O.  O.  Howard,  U.  S.  A. 

GENERAL  JACKSON.     By  James  Parton. 

GENERAL  GREENE.    By  Captain  Francis  V.  Greene,  U.  S.  A. 

GENERAL  J.  E.  JOHNSTON.    By  Robert  M.  Hughes,  of  Va. 

GENERAL   THOMAS.     By  Henry  Coppee,  LL.  D. 

GENERAL   SCOTT.     By  General  Marcus  J.  Wright. 

GENERAL  WASHINGTON.     By  Gen.  Bradley  T.  Johnson. 

GENERAL  LEE.     By  General  Fitzhugh  Lee. 

GENERAL  HANCOCK.     By  General  Francis  A.  Walker. 

GENERAL  SHERIDAN.     By  General  Henry  E.  Davies. 

These  are  volumes  of  especial  value  and  service  to  school  libraries,  either 
for  reference  or  for  supplementary  reading  in  history  classes.  Libraries, 
whether  public,  private,  or  school,  that  have  not  already  taken  necessary 
action,  should  at  once  place  upon  their  order-lists  the  Great  Commanders 
Series. 

The  following  are  in  press  or  in  preparation  : 

General  Sherman.     By  General  Manning  F.  Force. 

General  Grant.     By  General  James  Grant  Wilson. 

Admiral  Porter.     By  James  F.  Soley,  late  Assistant  Sec'y  of  Navy. 

General  McClellan.     By  General  Alexander  S.  Webh. 

General  Meade.    By  Richard  Meade  Bache. 

"  This  series  of  books  promises  much,  both  by  th°:  •  subjects  and  by  the  men  who 
have  undertal<en  to  write  them.  They  are  just  the  •  .ing  for  young  men  and  women ; 
delightful  reading  for  men  and  women  of  any  age."—  The  Evangelist, 

New  York  :   D.  APPLETON  &  CO.,  72  Fifth  Avenue. 


D.   APPLETON    &    CO.'S    PUBLICATIONS. 


H 


JOHN   BACH   MC  MASTER. 


ISTOR  Y  OF  THE  PEOPLE 

OF   THE    UNITED    STATES, 

from   the    Revolution    to   the   Civil 

War.    By  John  Bach  McMaster. 

To  be    completed   in   six   volumes. 

Vols.  I,  11,  III,  and  IV  now  ready. 

8vo.     Cloth,  gilt  top,  $2.50  each. 

"...  Prof.  !\IcMaster  has  tolti  us  what  no  other 
historians  have  told.  .  .  .  The  skill,  the  animation,  the 
brightness,  the  force,  and  the  charm  with  which  he  ar- 
rays the  facts  bef  )re  us  are  such  that  we  can  hardly 
conceive  of  more  interesting  reading  for  an  American 
citizen  who  cave.;  to  know  the  nature  of  those  causes 
which  have  made  not  only  him  but  his  environment 
and  the  opportunities  life  has  given  him  what  they  are." 
— N.  V.  Times. 


"Those  who  can  read  between  the  lines  may  discover  in  these  pages  constant 
evidences  of  care  and  skill  and  faithful  labor,  of  which  the  old- time  superficial  essay- 
ists, compiling  library  notes  on  dates  and  striking  events,  had  no  conception ;  but 
to  the  general  re.ider  the  fluent  narrative  gives  no  hint  of  the  conscientious  labors, 
far-reaching,  world-wide,  vast  and  yet  microscopically  minute,  that  give  the  strength 
and  value  which  are  felt  rather  than  seen.  This  is  due  to  the  art  of  presentation. 
The  author's  position  as  a  scientific  workman  we  may  accept  on  the  abundant  tes- 
ti:nony  of  the  experts  who  know  the  solid  worth  of  his  work ;  his  skill  as  a  literary 
artist  we  can  all  appreciate,  the  charm  of  his  style  being  self-evident." — Philadelphia 
Telegraph. 

•'The  third  volume  contains  the  brilliantly  written  and  fascinating  story  of  the  prog- 
ress and  doings  of  the  people  of  this  country  from  the  era  of  the  Louisiana  purchase 
to  the  opening  scenes  of  the  second  war  with  Great  Pritain— say  a  period  of  ten  years. 
In  every  page  of  the  book  the  reader  finds  that  fascinating  flow  of  narrative,  that 
clear  and  lucid  style,  and  that  penetrating  power  of  thought  and  judgment  which  dis- 
tinguished the  previous  volumes." — Columbus  State  JouruaL 

"  Prof  McMaster  has  more  than  fulfilled  the  promises  made  in  his  first  volumes, 
and  his  work  is  constantly  grosving  bet  :er  and  more  valuable  as  he  brings  it  nearer 
to  our  own  time.  His  style  is  clear,  simple,  and  idiomatic,  and  there  is  just  enough 
of  the  critical  spirit  in  the  narrative  to  guide  the  reader."— AW /r'W  Herald. 

"  Take  it  all  in  all,  the  History  promises  to  be  the  ideal  American  history.  Not  so 
much  given  to  dates  and  battles  and  great  events  as  in  the  fact  that  it  is  like  a  preat 
panorama  of  the  people,  revealing  their  iimer  life  and  action.  It  contains,  with  all  its 
boDcr  facts,  the  spice  of  personalities  and  incidents,  which  relieves  every  page  from 
dullness."— CVi/tvj^;^'!)  Inter-Ocean. 

"  History  written  in  this  picturesque  style  will  tempt  the  most  heedless  to  read. 
Prof.  McMaster  is  more  than  a  stylist;  he  is  a  student,  and  his  flistory  abounds  in 
evidences  of  research  in  quarters  not  before  discovered  by  the  historian." — Chicago 
Tribune. 


ture. 


"  A  History  sui  generis  which  has  made  and  will  keep  its  own  place  in  our  litera- 
3." — Netv  York  Evening  Post. 


"His  style  is  vigorous  and  his   treatment  candid  and   impartial."— AVw   Y'ork 
Tribune, 


New  York  :  D.  APPLETON  &  CO.,  72  Fifth  Avenue. 


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D.  APPLETON  &  CO.'S  PUBLICATIONS. 

CTUAL  AFRICA ;  or,  The  Coming  Continent.  A 
Tour  of  Exploration.  By  Frank  Vincent,  author  of  "The 
Land  of  the  White  Elephant,"  etc.  With  Map  and  io2  Illus- 
trations.    8vo.     Cloth,  $5.00. 

Tiiis  thorou2;h  and  comprehensive  work  furnishes  a  survey  of  the  entire  continent, 
Mvhich  this  experienced  traveler  has  circumnavigated  in  addition  to  his  inland  exploia- 
ti  ms.  The  latter  have  included  journeys  in  nortliein  Africa,  Madagascar,  southern 
Atrica,  and  an  expedition  into  tlie  Congo  country  which  has  covered  fresh  ground.  His 
boo  :  has  the  distinction  of  presenting  a  comprehensive  summary,  instead  of  offering  an 
account  of  one  special  district.  It  is  more  elaborately  illustrated  than  any  book  upon 
the  subject,  and  contains  a  large  map  carefully  corrected  to  date. 

"  Mr.  Frank  N'iiicent's  books  of  travel  meiit  to  be  ranked  among  the  very  best,  not 
on'y  for  their  thoroughness,  but  for  the  animation  of  their  narrative,  and  the  skill 
with  which  he  fastens  upon  his  reader's  mind  the  impression  made  upon  him  by  his 
voy agings." — Boston  Saturday  Evening  Gazette. 

"  A  new  volume  from  Mr.  P'rank  Vincent  is  always  welcome,  for  the  reading  public 
have  learned  to  regard  him  as  one  of  the  most  intelligent  and  observing  of  travelers." — 
New  York  Tribune. 


A 


ROUND  AND  ABOUT  SOUTH  AMERICA  : 

Twenty  Months  of  Quest  and  Qtieiy.     By  Frank  Vincent. 

With  Maps,  Plans,  and  54  full-page  Illustrations.     8vo,  xxiv  + 

473  p^ges.     Ornamental  cloth,  $5.00. 

"  South  America,  with  its  civilization,  its  resources,  and  its  charms,  is  being  con- 
stantly introduced  to  us,  and  as  constantly  surprises  us.  .  .  .  The  Parisian  who  thinks 
us  all  barbarians  is  probably  not  denser  in  his  prejudices  than  most  of  us  are  about  our 
Southern  continent.  We  are  content  not  10  know,  there  seeming  to  be  no  reason  why 
we  should.  Fashion  has  not  yet  directed  her  steps  there,  and  there  has  been  nothing 
to  stir  us  out  of  oar  lethargy.  .  .  .  Mr.  Vincent  observes  very  carefully,  is  always 
good-humored,  and  gives  us  the  best  of  what  he  sees.  .  .  .  The  reader  of  his  book  will 
gain  a  clear  idea  of  a  marvelous  country.  Maps  and  illustrations  add  greatly  to  the 
value  of  this  work." — New  York  Commercial  Advertiser. 

"The  author's  style  is  unusually  simple  and  straightforward,  the  printing  is  re- 
markably accurate,  and  the  forty-odJ  illustrations  are  refreshingly  original  for  the  most 
part." — The  Nation. 

"Mr.  Vincent  has  succeeded  in  giving  a  most  interesting  and  valuable  narmtive. 
His  account  is  made  d(nibly  valuable  by  the  exceptionally  good  illustiations,  most  of 
them  photographic  reproductions.  The  printing  of  both  text  and  plates  is  beyond 
criticism." — Philadelphia  Public  Ledger. 


I 


N  AND  OUT  OF  CENTRAL  AMERICA  ;  and 

other  Sketches  and  Studies  of  Travel.     By  Frank  Vincent. 
With  Maps  and  Illustrations.     i2mo.     Cloth,  $2.00. 

"  Few  living  travelers  have  had  a  literary  success  equal  to  Mr.  Vincent's."— 
Hiirfiers  H'eekly. 

"  Mr.  Vincent  has  now  seen  all  the  most  interesting  parts  of  the  world,  having 
traveled,  during  a  total  period  of  eleven  years,  two  hundred  and  sixty-five  thousand 
miles.  His  personal  knowledge  of  man  and  Nature  is  probably  as  varied  and  complete 
as  that  of  any  person  living  " — New  York  Home  Journal. 


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New  York :   D.  APPLETON  &  CO.,  72  Fifth  Avenue. 


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